Advances in Imaging and Electron Physics, Volume 149: Electron Emission Physics

ADVANCES IN IMAGING AND ELECTRON PHYSICS

VOLUME 149 ELECTRON EMISSION PHYSICS

EDITOR-IN-CHIEF

PETER W. HAWKES CEMES-CNRS Toulouse, France

HONORARY ASSOCIATE EDITORS

TOM MULVEY BENJAMIN KAZAN

Advances in

Imaging and Electron Physics Electron Emission Physics

BY

KEVIN L. JENSEN Electronics Science and Technology Division Naval Research Laboratory Washington, DC

VOLUME 149

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier

Academic Press is an imprint of Elsevier 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK

This book is printed on acid-free paper. Copyright ß 2007, Elsevier Inc. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the Publisher. The appearance of the code at the bottom of the first page of a chapter in this book indicates the Publisher’s consent that copies of the chapter may be made for personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc. (www.copyright.com), for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for pre-2007 chapters are as shown on the title pages. If no fee code appears on the title page, the copy fee is the same as for current chapters. 1076-5670/2007 $35.00 Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (þ44) 1865 843830, fax: (þ44) 1865 853333, E-mail: [email protected]. You may also complete your request on-line via the Elsevier homepage (http://elsevier.com), by selecting ‘‘Support & Contact’’ then ‘‘Copyright & Permission’’ and then ‘‘Obtaining Permissions.’’ For all information on all Elsevier Academic Press publications visit our Web site at www.books.elsevier.com ISBN: 978-0-12-374207-0 PRINTED IN THE UNITED STATES OF AMERICA 07 08 09 10 9 8 7 6 5 4 3 2 1

In memory of William D. Jensen (July 16, 1938 – July 4, 2007) for his inspirational devotion to science

This page intentionally left blank

CONTENTS

Dedication . . . . . . Preface . . . . . . . Future Contributions Foreword . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

v ix xi xvii

Field and Thermionic Emission Fundamentals . . . . . Thermal and Field Emission. . . . . . . . . . . . . Photoemission . . . . . . . . . . . . . . . . . . . Low–Work Function Coatings and Enhanced Emission . Appendices . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

4 47 147 280 304 306 309

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

325

Electron Emission Physics

Kevin L. Jensen I. II. III. IV. V. VI.

vii

This page intentionally left blank

PREFACE Electron emission physics is too vast a theme to be confined within a regular review article. In this volume, Kevin Jensen examines numerous aspects of the subject, in particular those of importance in recent generations of the related devices. A first long section recapitulates the fundamentals and serves as an introduction to the three succeeding sections. The second covers the mechanisms of thermal and field emission; the various models are described and expressions for current density and related quantities are derived in the two extreme cases. A valuable feature of this chapter is the meticulous examination of the approximations involved, always a source of debate. All the steps in the relatively complicated derivations are shown. Next comes a long section on photoemission with, as before, a presentation of the models used and the associated physics, culminating in a study of the emittance and brightness of photocathodes. A last section, very much the physics of electron emission, discusses coatings with materials of low work-function and the resulting increase in emission. This monograph undoubtedly fills a gap in the literature, and I am delighted that it should appear in these Advances. I shall not be alone in appreciating the eVort made to present all this material so clearly. Peter W. Hawkes

ix

This page intentionally left blank

FUTURE CONTRIBUTIONS

S. Ando Gradient operators and edge and corner detection P. Batson (special volume on aberration-corrected electron microscopy) Some applications of aberration-corrected electron microscopy C. Beeli Structure and microscopy of quasicrystals A. B. Bleloch (special volume on aberration-corrected electron microscopy) Aberration correction and the SuperSTEM project C. Bontus and T. Ko¨hler (vol. 151) Reconstruction algorithms for computed tomography G. Borgefors Distance transforms Z. Bouchal Non-diVracting optical beams A. Buchau Boundary element or integral equation methods for static and timedependent problems B. Buchberger Gro¨bner bases L. Busin, N. Vandenbroucke, and L. Macaire (vol. 151) Color spaces and image segmentation G. R. Easley and F. Colonna Generalized discrete Radon transforms and applications to image processing T. Cremer Neutron microscopy I. Daubechies, G. Teschke, and L. Vese (vol. 150) On some iterative concepts for image restoration

xi

xii

FUTURE CONTRIBUTIONS

A. X. Falca˜o The image foresting transform R. G. Forbes Liquid metal ion sources C. Fredembach Eigenregions for image classification A. Go¨lzha¨user Recent advances in electron holography with point sources D. Greenfield and M. Monastyrskii Selected problems of computational charged particle optics M. Haider (special volume on aberration-corrected electron microscopy) Aberration correction in electron microscopy M. I. Herrera The development of electron microscopy in Spain N. S. T. Hirata Stack filter design M. Hy¨tch, E. Snoeck, and F. Houdellier (special volume on aberrationcorrected electron microscopy) Aberration correction in practice K. Ishizuka Contrast transfer and crystal images J. Isenberg Imaging IR-techniques for the characterization of solar cells A. Jacobo Intracavity type II second-harmonic generation for image processing B. Kabius (special volume on aberration-corrected electron microscopy) Aberration-corrected electron microscopes and the TEAM project L. Kipp Photon sieves A. Kirkland and P. D. Nellist (special volume on aberration-corrected electron microscopy) Aberration-corrected electron micrsocpy

FUTURE CONTRIBUTIONS

xiii

G. Ko¨gel Positron microscopy T. Kohashi Spin-polarized scanning electron microscopy O. L. Krivanek (special volume on aberration-corrected electron microscopy) Aberration correction and STEM R. Leitgeb Fourier domain and time domain optical coherence tomography B. Lencova´ Modern developments in electron optical calculations H. Lichte New developments in electron holography M. Matsuya Calculation of aberration coeYcients using Lie algebra S. McVitie Microscopy of magnetic specimens S. Morfu and P. Marquie´ Nonlinear systems for image processing T. Nitta Back-propagation and complex-valued neurons M. A. O’Keefe Electron image simulation D. Oulton and H. Owens Colorimetric imaging N. Papamarkos and A. Kesidis The inverse Hough transform R. F. W. Pease (vol. 150) Significant advances in scanning electron microscopy, 1965–2007 K. S. Pedersen, A. Lee, and M. Nielsen The scale-space properties of natural images S. J. Pennycook (special volume on aberration-corrected electron microscopy) Some applications of aberration-corrected electron microscopy

xiv

FUTURE CONTRIBUTIONS

E. Plies (special volume on aberration-corrected electron microscopy) Electron monochromators T. Radlicˆka (vol. 151) Lie algebraic methods in charged particle optics V. Randle (vol. 151) Electron back-scatter diVraction E. Rau Energy analysers for electron microscopes E. Recami Superluminal solutions to wave equations J. Rodenburg (vol. 150) Ptychography and related diVractive imaging methods H. Rose (special volume on aberration-corrected electron microscopy) The history of aberration correction in electron microscopy G. Schmahl X-ray microscopy J. Serra (vol. 150) New aspects of mathematical morphology R. Shimizu, T. Ikuta, and Y. Takai Defocus image modulation processing in real time S. Shirai CRT gun design methods T. Soma Focus-deflection systems and their applications J.-L. Starck Independent component analysis: the sparsity revolution I. Talmon Study of complex fluids by transmission electron microscopy N. Tanaka (special volume on aberration-corrected electron microscopy) Aberration-corrected microscopy in Japan M. E. Testorf and M. Fiddy Imaging from scattered electromagnetic fields, investigations into an unsolved problem

FUTURE CONTRIBUTIONS

xv

N. M. Towghi Ip norm optimal filters E. Twerdowski Defocused acoustic transmission microscopy Y. Uchikawa Electron gun optics K. Urban and J. Mayer (special volume on aberration-corrected electron microscopy) Aberration correction in practice K. Vaeth and G. Rajeswaran Organic light-emitting arrays M. van Droogenbroeck and M. Buckley Anchors in mathematical morphology R. Withers Disorder, structured diVuse scattering and local crystal chemistry M. Yavor Optics of charged particle analysers Y. Zhu (special volume on aberration-corrected electron microscopy) Some applications of aberration-corrected electron microscopy

This page intentionally left blank

FOREWORD

There is much to the observation of J. M. Ziman (2001), an exceptionally clear translator of the whisperings of the tenth Muse, when he noted in his preface to Electrons and Phonons that ‘‘Like a chemical compound, scientific knowledge is purified by recrystallization,’’ followed by several more breathtaking metaphors about the value of distilling hard‐won scientific insights into texts. The debt of the present effort to him lurks behind many a page written here, giving credence to his insight. After my having profited enormously from the hard‐won nucleations of previous generations, it is time to contribute in turn. There is much merit in the international literature on electron emission physics. To do justice to the field in a short work—or to even read what is there, much less distill it—is daunting. Present aims perforce are much more modest. Recognizing that a representation of what exists cannot be adequately conveyed to those who wish to look, I shall instead try to convey what I saw when I looked, along with travel notes of the journey (which describes some features of the process—‘‘random walk’’ describing the others). To the many whose work has been ignored by such an itinerary, my intent is not to slight by omission of discussion or reference to meritorious work. The whole process of getting a simple electron from inside a material into a vacuum cuts across many disciplines in physics, and it is therefore no surprise that many renowned names appear, often repeatedly, from the early decades of the twentieth century. If not for the ‘‘physicists’ war,’’ as World War II has come to be called in some circles (see, for example, Chapter 20 of Kevles, 1987), perhaps some of the great names of physics that are reverently mentioned herein would not be so widely appreciated outside the high walls of academia. But greatness is not something that is only born of conflict. Indeed, progress in physics is largely due to international collegiality, open discussion, much input from colleagues, and mentorship. I have had the pleasure of association with many whom I hold in high regard. My experience, such as it is, is that physics only looks magisterial in the foundation myths where goateed graybeards pontificate from podiums. Physics research is a gritty, wonderful struggle, and the give and take, the clashing of ideas, the absence of certainty, make for very powerful and compelling theater where the boundary between actor and audience is gone. I am grateful for the honor and pleasure of sharing the stage with many colleagues. I have tried to give xvii

xviii

FOREWORD

some of them their due here where possible, perhaps imperfectly. I would like to thank some by name, although there are many more to whom I am grateful (they know who they are). It is a sublime feature of physics that the enterprise is far greater than its practitioners, traceable to progress being a collective effort. Still, what defects exist herein are mine and do not reflect on those whom I call colleagues and friends. I have had the distinct pleasure of learning a great deal from my colleagues at the Naval Research Laboratory over the years: F. A. Buot, J. Calame, H. Freund, A. Ganguly, M. A. Kodis, Y. Y. Lau, B. Levush, K. Nguyen, P. Phillips, T. Reinecke, J. L. Shaw, A. Shih, J. E. Yater, and E. G. Zaidman. A research environment in which expertise is but a walk down the hall or near a coffee pot has no equal. In 2001, I had the distinct pleasure of spending a sabbatical at the University of Maryland and since then have enjoyed my weekly visits. My UMD colleagues have been open, gregarious, stimulating, and beneficial: P. G. O’Shea, D. W. and R. Feldman, N. A. Moody (now at Los Alamos National Laboratory), D. Demske, and E. Montgomery. I remain deeply indebted to P. O’Shea and D. Feldman for encouraging interesting problems at serendipitous moments. I would like to thank the Feldmans in particular for sharing their international friendships simply because of an idle dinner conversation remark that has allowed me to pursue something I have long dreamed of doing—namely, this. There have been many whose camaraderie, insight, and/or guidance have been invaluable, some of whom are T. Akinwande, S. Bandy, I. Ben‐Zvi, S. Biedron, V. T. Binh, C. A. Brau, I. Brodie, H. Busta, F. Charbonnier, W. B. Colson, P. Cutler, D. H. Dowell, R. G. Forbes, B. E. Gilchrist, M. C. Green, C. Holland, M. A. Hollis, C. Hunt, J. W. Lewellen, L. G. Il’chenko, R. T. Longo, W. A. Mackie, C. Marrese‐Reading, R. A. Murphy, R. Nemanich, G. Nolting, W. D. Palmer, J. K. Percus, J. J. Petillo, T. Rao, Q. Saulter, P. R. Schwoebel, J. Severns, J. M. Smedley, T. Smith, D. Temple, A. Todd, R. J. Umstattd, E. G. Wintuckey, W. Zhu, and J. D. Zuber. I have particularly enjoyed the many occasions I have spent with C. A. (Capp) Spindt, who has always been gracious, a good friend, and a pleasurable colleague. I wish to honor the memory of three people, each of whom has left their unique mark on me during my tenure: H. F. Gray, R. K. Parker, and C. Bohn. They shall always live on in their work, but so, too, in my recollections of my time with them. I owe considerable gratitude to the Naval Research Laboratory for its many years of support, for the broad education I was able to pursue during my tenure there, and for indulging my brand of basic research. What I have to give was made possible through their investments in me, particularly

FOREWORD

xix

while R. K. Parker was at the helm of the Vacuum Electronics Branch. I also thank the Office of Naval Research and the Joint Technology Office for their support over the years. I thank Peter Hawkes for his great patience, for making possible this wonderful opportunity, and for his efforts to make its realization good, hopefully as good as the dream; and Tracy Grace for the difficult task of dampening stochastic thought into coherent narrative. They did so with much humor and poise. To my children, who keep me young, but who have first made me old—and I hope much wiser. And to my parents, who raised me to hold the passions and ethics I do. I’ve never regretted following their footsteps. And to my wife, whose centrality especially in uncertain times was never in doubt. I owe much to three generations of women in my life: grandmother, mother, and wife, each of whom has bequeathed their own special gifts to me. The Bard spoke truly: ‘‘From women’s eyes this doctrine I derive: / They are the ground, the books, the academes, / From whence doth spring the true Promethean fire’’ (Shakespeare, Love’s Labor’s Lost, Act 4, Scene III). Thank you, thank you, thank you. Kevin L. Jensen

ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 149

Electron Emission Physics KEVIN L. JENSEN

I. Field and Thermionic Emission Fundamentals . . . . . . . . . . A. A Note on Units . . . . . . . . . . . . . . . . . . . B. Free Electron Gas . . . . . . . . . . . . . . . . . . . 1. Quantum Statistical Mechanics. . . . . . . . . . . . . . 2. The Fermi–Dirac Integral . . . . . . . . . . . . . . . 3. The Chemical Potential . . . . . . . . . . . . . . . . 4. A Phase Space Description . . . . . . . . . . . . . . . C. Nearly Free Electron Gas . . . . . . . . . . . . . . . . 1. The Hydrogen Atom . . . . . . . . . . . . . . . . . 2. Band Structure and the Kronig–Penney Model . . . . . . . . 3. Semiconductors . . . . . . . . . . . . . . . . . . . 4. Band Bending . . . . . . . . . . . . . . . . . . . D. The Surface Barrier to Electron Emission . . . . . . . . . . . 1. Surface Effects and Origins of the Work Function . . . . . . . 2. Ion Core Effects . . . . . . . . . . . . . . . . . . . 3. Dipole Effects Due to Surface Barriers . . . . . . . . . . . E. The Image Charge Approximation . . . . . . . . . . . . . 1. Classical Treatment . . . . . . . . . . . . . . . . . 2. Quantum Mechanical Treatment . . . . . . . . . . . . . 3. An ‘‘Analytical’’ Image Charge Potential . . . . . . . . . . II. Thermal and Field Emission . . . . . . . . . . . . . . . . . A. Current Density . . . . . . . . . . . . . . . . . . . . 1. Current Density in the Classical Distribution Function Approach . . 2. Current Density in the Schro¨dinger and Heisenberg Representations . 3. Current Density in the Wigner Distribution Function Approach . . 4. Current Density in the Bohm Approach . . . . . . . . . . B. Exactly Solvable Models . . . . . . . . . . . . . . . . . 1. Wave Function Methodology for Constant Potential Segments. . . 2. The Square Barrier . . . . . . . . . . . . . . . . . . 3. Multiple Square Barriers . . . . . . . . . . . . . . . . 4. The Airy Function Approach . . . . . . . . . . . . . . 5. The Triangular Barrier . . . . . . . . . . . . . . . . C. Wentzel–Kramers–Brillouin WKB Area Under the Curve Models . . 1. The Quadratic Barrier . . . . . . . . . . . . . . . . . 2. The Image Charge Barrier . . . . . . . . . . . . . . . D. Numerical Methods . . . . . . . . . . . . . . . . . . 1. Numerical Treatment of Quadratic Potential . . . . . . . . . 2. Numerical Treatment of Image Charge Potential . . . . . . . 3. Resonant Tunneling: A Numerical Example . . . . . . . . .

ISSN 1076-5670/07 DOI: 10.1016/S1076-5670(07)49001-2

1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 4 5 5 8 9 11 11 11 13 20 20 22 22 31 33 40 40 42 43 47 47 47 49 52 62 65 65 67 69 71 80 85 85 87 94 95 95 99

Copyright 2007, Elsevier Inc. All rights reserved.

2

KEVIN L. JENSEN

E. The Thermal and Field Emission Equation . . . . . . . . . 1. The Fowler–Nordheim and Richardson–Laue–Dushman Equations 2. The Emission Equation Integrals and Their Approximation . . . 3. The Revised FN and RLD . . . . . . . . . . . . . . F. The Revised FN‐RLD Equation and the Inference of Work Function From Experimental Data . . . . . . . . . . 1. Field Emission . . . . . . . . . . . . . . . . . . 2. Thermionic Emission . . . . . . . . . . . . . . . . 3. Mixed Thermal‐Field Conditions . . . . . . . . . . . . 4. Slope‐Intercept Methods Applied to Field Emission . . . . . G. Recent Revisions of the Standard Thermal and Field Models . . . 1. The Forbes Approach to the Evaluation of the Elliptical Integrals 2. Emission in the Thermal‐Field Transition Region Revisited . . . H. The General Thermal‐Field Equation . . . . . . . . . . . I. Thermal Emittance. . . . . . . . . . . . . . . . . . III. Photoemission . . . . . . . . . . . . . . . . . . . . A. Background . . . . . . . . . . . . . . . . . . . . B. Quantum Efficiency . . . . . . . . . . . . . . . . . C. The Probability of Emission . . . . . . . . . . . . . . 1. The Escape Cone . . . . . . . . . . . . . . . . . 2. The Fowler–Dubridge Model . . . . . . . . . . . . . D. Reflection and Penetration Depth . . . . . . . . . . . . 1. Dielectric Constant, Index of Refraction, and Reflectivity . . . 2. Drude Model: Classical Approach . . . . . . . . . . . 3. Drude Model: Distribution Function Approach . . . . . . . 4. Quantum Extension and Resonance Frequencies . . . . . . E. Conductivity . . . . . . . . . . . . . . . . . . . . 1. Electrical Conductivity . . . . . . . . . . . . . . . 2. Thermal Conductivity . . . . . . . . . . . . . . . . 3. Wiedemann–Franz Law . . . . . . . . . . . . . . . 4. Specific Heat of Solids. . . . . . . . . . . . . . . . F. Scattering Rates. . . . . . . . . . . . . . . . . . . 1. Fermi’s Golden Rule . . . . . . . . . . . . . . . . 2. Charged Impurity Relaxation Time . . . . . . . . . . . 3. Electron-Electron Scattering . . . . . . . . . . . . . 4. A Sinusoidal Potential. . . . . . . . . . . . . . . . 5. Monatomic Linear Chain of Atoms . . . . . . . . . . . 6. Electron-Phonon Scattering . . . . . . . . . . . . . . 7. Matthiesen’s Rule and the Specification of Scattering Terms . . G. Scattering Factor . . . . . . . . . . . . . . . . . . H. Temperature of a Laser-Illuminated Surface . . . . . . . . . 1. Photocathodes and Drive Lasers . . . . . . . . . . . . 2. A Simple Model of Temperature Increase Due to a Laser Pulse . 3. Diffusion of Heat and Corresponding Temperature Rise . . . . 4. Multiple Pulses and Temperature Rise . . . . . . . . . . 5. Temperature Rise in a Single Pulse: The Coupled Heat Equations. 6. The Electron-Phonon Coupling Factor g: A Simple Model . . .

. . . .

. . . .

. . . .

. . . .

102 104 106 110

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

118 118 121 123 127 131 131 136 139 143 147 147 148 151 151 152 154 154 156 158 162 165 165 167 170 171 174 174 177 180 185 186 194 212 215 222 222 223 225 227 234 236

3

ELECTRON EMISSION PHYSICS I. Numerical Solution of the Coupled Thermal Equations . . . . . 1. Nature of the Problem. . . . . . . . . . . . . . . . 2. Explicit and Implicit Solutions of Ordinary Differential Equations 3. Numerically Solving the Coupled Temperature Equations With Temperature-Dependent Coefficients. . . . . . . . . . . J. Revisions to the Modified Fowler–Dubridge Model: Quantum Effects K. Quantum Efficiency Revisited: A Moments-Based Approach . . . L. The Quantum Efficiency of Bare Metals . . . . . . . . . . 1. Variation of Work Function With Crystal Face . . . . . . . 2. The Density of States With Respect to the Nearly Free Electron Gas Model. . . . . . . . . . . . . . . . . . . . 3. Surface Structure, Multiple Reflections, and Field Enhancement . 4. Contamination and Effective Emission Area . . . . . . . . M. The Emittance and Brightness of Photocathodes . . . . . . . IV. Low–Work‐Function Coatings and Enhanced Emission . . . . . . A. Historical Perspective . . . . . . . . . . . . . . . . . B. A Simple Model of a Low–Work‐Function Coating . . . . . . C. A Less Simple Model of the Low–Work‐Function Coating . . . . D. The (Modified) Gyftopoulos–Levine Model of Work Function Reduction . . . . . . . . . . . . . . . . . E. Comparison of the Modified Gyftopoulos–Levine Model to Thermionic Data . . . . . . . . . . . . . . . . . . F. Comparison of the Modified Gyftopoulos–Levine Model to Photoemission Data . . . . . . . . . . . . . . . . . V. Appendices. . . . . . . . . . . . . . . . . . . . . . A. Integrals Related to Fermi–Dirac and Bose–Einstein Statistics . . . B. The Riemann Zeta Function . . . . . . . . . . . . . . VI. Conclusion . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

239 239 240

. . . . .

. . . . .

. . . . .

. . . . .

247 253 255 260 261

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

264 264 269 274 280 280 281 282

. . . .

286

. . . .

292

. . . . . .

296 304 304 305 306 309

How can my Muse want subject to invent, While thou dost breathe that pour’st into my verse Thine own sweet argument, too excellent, For every vulgar paper to rehearse?, O give thy self the thanks, if aught in me Worthy perusal stand against thy sight, For who’s so dumb that cannot write to thee, When thou thy self dost give invention light? Be thou the tenth Muse, ten times more in worth Than those old nine which rhymers invocate, And he that calls on thee, let him bring forth Eternal numbers to outlive long date. If my slight Muse do please these curious days, The pain be mine, but thine shall be the praise. Sonnet 38, William Shakespeare

. . . . . .

. . . . . .

. . . . . .

4

KEVIN L. JENSEN

I. FIELD AND THERMIONIC EMISSION FUNDAMENTALS A. A Note on Units The widespread application of electron source technology as a subdiscipline of physics and engineering disciplines is beholden to the use of SI (International System of Units) (meter‐kilogram‐second‐ampere [MKSA]) in formulas useful to experimenters. Despite its practical value, such a yoke is not always easy. For electron emission from nanoscale sites, SI units necessitate bookkeeping of inconveniently large exponents. The description of emission phenomena often finds units comparable to those of the Bohr atom (the sine qua non of the physicist’s lexicon) to be in play, for which scales of energy, distance, and charge are naturally introduced and described by electron volts, nanometers, femtoseconds, electron charge, and Kelvin (eV‐nm‐fs‐ q‐K) and are often used here alongside SI units. The waning unit of Angstrom, which occasionally appears, seems an odd choice, but it, along with the use of electron volt for energy, is commonly used in surface physics and emission phenomena. The indolent convention of q ¼
h ¼ c ¼ m ¼ 1, adopted when the relation of theory to experiment is not pressing or when obfuscation is useful, is shunned. Tables 1 and 2 summarize common relationships and conversions. Particularly important is how the electron charge is handled. The work function and electron affinity of metals and semiconductors is generally expressed in electron volts. Thus, rather than deal with electron charge, potentials, and fields separately, it is inordinately convenient to combine the unit charge with potential to get energy (eV) and with field to get force (eV/nm). Moreover, equations concerning potential (e.g., Poisson’s equation) are easily related to those concerning energy (e.g., Schro¨dinger’s equation) if the product of unit charge and volt are combined; if the charge of the electron is the unit used, then charge density and current are interchangeable with number density TABLE 1 FUNDAMENTAL CONSTANTS Quantity

Symbol

MKSA

˚ ‐fs‐q eV‐A

Bohr radius Electron rest energy Rydberg energy Permittivity of free space Planck’s constant Speed of light in vacuum Fine structure constant

ao mc2 Ry
o h
c afs

0.529177
10–10m 8.1871
10–31 J 2.17987
10–18 J 8.85419
10–12 C/Vm 1.05457
10–34 J s 2.997924
108 m/s 1/137.036

˚ 0.529177 A 510999 eV 13.6060 eV ˚ eV 5.52635x10–3 q2/A 0.658212 eV fs 2997.924 1/137.036

MKSA, meter‐kilogram‐second‐ampere.

ELECTRON EMISSION PHYSICS

5

TABLE 2 RELATION OF NANO UNITS TO SI* Quantity

˚ fq eA

Conversion factor

SI (MKSA)

Charge Length Time Energy Current Current density Density Field Energy Potential Resistance Permitivity

q ˚ A fs eV q/fs ˚2 q/fs A 3 ˚ q/A ˚ eV/q A

1.60218
1019 1010 1015 1.60218
1019 1.60218
104 1.60218
1012 160218 1010 1.60218
1019 1 1.60218
1014 1.60218
109

Coulomb meter second joule amp amp/cm2 Coulomb/cm3 volt/meter joule volt ohm Farad/meter

eV eV/q eV fs/q2 ˚ q2/eV A

MKSA, meter‐kilogram‐second‐ampere. ˚ fq) units in terms of MKSA, multiply MKSA by the conversion factor; MKSA *To obtain (eA ˚ fq) is given by the inverse of the conversion factor; e.g., for current density, units in terms of (eA ˚ 2) ¼ 1.60218
1012 A/cm2. The units in the MKSA column are those often used in (q/fs A practice, as in A/cm2 for current density.

and current. The convention used here is to combine potentials and fields with unit charge q so they become potential energy V [eV] and force F [eV/nm], respectively. A particularly useful related unit is the product of the fine structure constant, Planck’s constant, and the speed of light, or Q ¼ afs
hc=4 ¼ 0.359991 eV‐nm ¼ q2/16peo. Q appears frequently in the discussion of the image charge contribution to the potential in vacuum, for which the classical image charge potential energy is Q/x, x being the distance from the surface. B. Free Electron Gas 1. Quantum Statistical Mechanics The energy and the density of a gas of electrons permeates the discussion of the physics of electron emission, and it is therefore only fitting to explore them in the requisite detail. Consider a box of N (spinless) particles with total energy E. If the energy is parabolic in momentum, (which will be assumed henceforth), then energy levels are characterized by Ek ¼ ð
hkÞ2 =2m, where k is the vector corresponding to momentum. In a cubic box, the momentum is quantized as per ^ þ ky ^ k ¼ kx x y þ kz^z; kx ¼ plx =L; ky ¼ ply =L; kz ¼ plz =L

ð1Þ

6

KEVIN L. JENSEN

where l is an integer and V ¼ L3. The subscript k on E is not bold, as the energy depends only on the magnitude of the momentum. Particles of the same energy are grouped into levels characterized by an energy Ei. A state consists of ni particles distributed among gi levels. Consequently, the total particle number and energy for the system are given by P P N ¼ Pk nk ¼ iP ni ; ð2Þ E ¼ k nk Ek ¼ i ni Ei where the first sum is a sum over quantum numbers (i.e., nk is an occupation number) and the second a sum over levels (i.e., ni is the sum over all nk characterized by energy Ei). Define W fni g as the number of states of the box corresponding to the set of occupation numbers fni g. The entropy of the system is given by S ¼ kB ln ðW fni gÞ;

ð3Þ

where kB is Boltzmann’s constant. Isolated systems in equilibrium are in a state of maximum entropy; that is, fluctuations will cause a decrease in S if the system is in equilibrium. The state variables are given by particle number N, volume V (recall the definition of Ek), and entropy S for systems in thermal and mechanical contact with the outside (Reichl, 1987). Changes in energy are therefore related to the state variables by dE ¼ TdS  PdV þ mdN;

ð4Þ

where m is the ‘‘chemical potential’’—which is therefore seen as the change in energy when the number of particles is increased. If wi is the number of ways in which ni particles can be allocated to the gi locations with a cell, then it follows that X ln ðwi Þ; ð5Þ lnðW fni gÞ ¼ i where wi is deduced from counting arguments. The entropy of a system is the sum of the entropies of the subsystems, and so Si ¼ kB lnðwi Þ. The ‘‘statistics’’ of the particles is crucial in the understanding of emission current, for example, and so it is profitable to concentrate on the meaning of the designation. a. Maxwell–Boltzmann Statistics. For Maxwell–Boltzmann (MB) statistics, there are N! ways to place N particles into different levels, but if the particles are indistinguishable, then there are only N!=Pi ðni !Þ distinct arrangements. Within each level, each particle can be placed in gi locations, so ni particles will each separately contribute a factor of gi to the combinatorics. In order that wi so defined is the asymptotic limit of the Fermi–Dirac

ELECTRON EMISSION PHYSICS

7

(FD) and Bose–Einstein (BE) distributions, wi is divided by N! (correct Boltzmann counting) and so (Leonard and Martin, 1980) wi jMB ¼

X g ni i : i n!

ð6Þ

b. Fermi–Dirac Statistics. There are (gi) locations to place the first particle within a level. The Pauli exclusion principle restricts the occupation number of each momentum state to be 0 or 1, so there are but ðgi  1Þ locations for the next particle, and so on until the ni‐th particle. As with the MB case, a factor of ni! accounts for indistinguishable permutations within a level, and so (accounting for spin‐1/2 particles will square each term in the sum) X 1 Yni 1 X 1
gi !  wi jFD ¼ ð gi  k Þ ¼ : ð7Þ in! i n ! ðg  n Þ! k¼0 i i i i c. Bose–Einstein Statistics. For bosons, there is no restriction on the number of particles that can occupy a given momentum state. The number of permutations of the ni particles and the ðgi  1Þ partitions must both be accounted for, and so X 1
ðni þ gi  1Þ! wi jBE ¼ ð8Þ in! ðgi  1Þ! i From Eqs. (6)–(8), both BE and FD statistics approximate MB statistics if gi  ni , that is, the number of particles in each level is small compared with the number of available locations, a circumstance characteristic of high temperature. Invoking Stirling’s approximation lnðn!Þ  n lnðnÞ  n and neglecting terms < O(1/n), the subsystem entropies satisfy   @ @ gi Si ¼ kB lnðwi Þ ¼ kB ln s ; ð9Þ @ni @ni ni where s ¼ f1; 0; 1g for FD, MB, and BE statistics, respectively. Maximizing the entropy S subject to the constraints of Eq. (2) is equivalent to finding the ni for which hP   i P  P S þ a N  n n E 0 ¼ @ ni þ b E  j j j j j j j ð10Þ ¼ @ ni Si  a  bEi where a and b are undetermined multipliers. From Eqs. (9) and (10), it follows that for each level, ni ðEi Þ ¼ gi =½s þ expða þ bEi Þ and therefore, for each momentum vector k nðEk Þ ¼ ½s þ expða þ bEk Þ1 :

ð11Þ

8

KEVIN L. JENSEN

To find a and b, combine the derivative of Eq. (2) with Eq. (10) to obtain P dE ¼ i ð0ni dEi þ1Ei dni Þ P @Ei A 1 X aX ð12Þ dV þ ¼ i @ ni dSi  dni i i kB b b @V where the sums over dSi and dni give dS and dN, respectively. Comparing the coefficients of dS and dN in Eq. (12) with Eq. (4) identifies b ¼ 1=kB T and a ¼ m=kB T and therefore nðEk Þ ¼ fs þ exp½bðEk  mÞg1 :

ð13Þ

The sum of Eq. (13) over all momentum states, as per Eq. (2), gives the total number of particles N. In the continuum limit for fermions (s ¼ 1), and including a factor of 2 to account for the spin‐1/2 nature of electrons, N¼



X

! nðkÞ k

)2

L 2p

3 ð

dk : ½1 þ exp½bðEðkÞ  mÞ

ð14Þ

The chemical potential m was treated as an inauspicious parameter, but it is of central significance and is the derivative of the free energy with respect to the occupation number. In the free electron model for a box of volume L3, the energy is given by E ðkÞ ¼


2 p2  2 h 2 2 l þ l þ l y z ¼ EðkÞ: 2mL2 x

ð15Þ

In the (zero temperature) ground state, electrons are added until each level is filled to its maximum capacity; the momentum of the last electron in is the Fermi momentum
hkF . The chemical potential is identified with the corresponding Fermi energy.

2. The Fermi–Dirac Integral Introducing the number density r ¼ N/V. E(k) depends on the magnitude of k, so that in spherical coordinates dk ¼ 4p k2 dk, Eq. (14) becomes  3=2 4 m rðm,T Þ ¼ pffiffiffi F1=2 ðbmÞ; p 2pb
h2

ð16Þ

9

ELECTRON EMISSION PHYSICS

where the FD integral of order p, denoted Fp(x), is defined by 1 ð

Fp ðxÞ ¼ 0

yp dy: 1 þ eyx

ð17Þ

Blakemore (1987) provides a general discussion and tables of Fermi–Dirac integrals of order p. For negative argument and p ¼ 1/2 pffiffiffi X p 1 3=2 F1=2 ðx < 0Þ ¼ n ð1Þn þ 1 enx 2 n¼1 0 2 ð18Þ pffiffiffi1 31 pffiffiffi ex 1 p x4 3A 2x 5 @  e 1 þ pffiffiffi þ e  8 2 9 2 2 where the second line is good to better than 1% for x  0.2. For positive argument (Jensen and Ganguly, 1993) 8 9 1 pffiffiffiffiffiffiffiffiffiffiffi ð pffiffiffiffiffiffiffiffiffiffiffi = <2 ð 1 pffiffiffiffiffiffiffiffiffiffiffi 1 þ y  1  y 1 þ y þ dy þ dy : ð19Þ F1=2 ðx > 0Þ ¼ x3=2 :3 exy þ 1 exy þ 1 ; 0 1

For x  1, the last integral can be ignored. Taylor expanding the radicals in the middle integral and taking the upper limit to (þ1) results in terms proportional to the Riemann zeta function z(2n) (see Appendix 1). A reasonable approximation for x  2.5 is 8 9 1 <2 X  zð2n þ 2Þ= ð4nÞ!

F1=2 ðx  1Þ ¼ x3=2 þ 1  22n1 4n :3 n ¼ 0 2 ð2nÞ! x2nþ2 ; ð20Þ 0 12 0 14 31 2 2 3=2 4 1@ p A 3 @ p A5  x 1 þ 3 2 2x 40 2x For intermediate values of x, a quadratic approximation with an error of less than 1% is F1=2 ð0:2  x  2:5Þ  0:1897x2 þ 0:5362x þ 0:6705 ð21Þ The performance of the approximations in Eqs. (18)–(21) is shown in Figure 1. 3. The Chemical Potential At room temperature, the coefficient of F1=2 ðbmÞ in Eq. (16) is 2.832
1019 #/cm3. In the ‘‘free electron Fermi gas’’ model (Kittel, 1996), the electron number density is approximately the same as the atomic number density and

10

KEVIN L. JENSEN

10 [20]

1.0 0.5 0.0

1

% Error

F1/2(x)

[21]

−0.5 −1.0

[18] 0.1 −2

−1

0

1

2

3

4

x FIGURE 1. The Fermi–Dirac integral (circles) compared to the approximations (lines) and the associated error (dashed lines).

on the order of 0.1 moles/cm3, or three orders of magnitude larger than the coefficient. It is clear, therefore, that bm is generally large and positive for metals, and Eq. (20) is a good approximation. For semiconductors, the mass m is interpreted as the peffective electron ffiffiffi mass m*. The coefficient of F1=2 ðbmÞ is designated NC ð2= pÞ, where NC is the ‘‘effective density of conduction band states’’ (an analogous equation exists for the valence band). The number of conduction band electrons in an n‐type semiconductor is a temperature‐dependent fraction of the dopant concentration. For silicon, generic doping concentrations are 1015 to 1018 #/cm3, indicating that the chemical potential is negative and that Eq. (18) holds. The number density r does not vary with regard to temperature; therefore, the chemical potential is temperature dependent in such a way as to offset the temperature‐dependence of NC. The ‘‘Fermi level’’ EF ¼ mo is taken as mðT ¼ 0K Þ  mo ¼ ð
hkF Þ2 =2m, where
hkF is the Fermi momentum. In the zero temperature limit  3=2 4 m 2 k3 ðbmo Þ3=2 ¼ F2 ð22Þ rðmo ; 0Þ ¼ lim pffiffiffi 2 b!1 p 2pb
3 3p h For metals, the temperature dependence of m(T) is obtained by setting r(mo,0) ¼ r(m,T ) and using Eq. (20) to derive     ! 1 pkB T 2 1 pkB T 4  mðT Þ  mo 1  ð23Þ 3 2mo 5 2mo

ELECTRON EMISSION PHYSICS

11

For a generic metallic density of 0.1 mole/cm3, mo ¼ 5.6023 eV and ˚ –1; even at 3000 K, m(T) is within 99.8% of mo. Consequently, kF ¼ 1.2126 A the temperature dependence of m is often neglected and m is taken as interchangeable with EF for metals. 4. A Phase Space Description The generalization of the Fermi distribution is a phase space distribution f ðr,kÞ such that f ðr,kÞd 3 rd 3 k is the number of particles in the phase space element d 3 rd 3 k. The Boltzmann transport equation describes the evolution of the distribution function: from the conservation of the distribution along a flow line, that is, f ðr; k; tÞ ¼ f ðr þ dr,k þ dk; t þ dtÞ, which implies 

   ! @ F ! @f þ v r r þ r k f ðr,k,tÞ ¼ , @t h
@t coll

ð24Þ

where v and F=m are the velocity and acceleration, respectively (recall that F is the product of the electron charge and electric field,) and the right‐hand side (RHS) represents the effects of collisions and scattering on the distribution. For the steady‐state case, and neglecting the collision operator, Eqs. (13) and (24) indicate that the electrochemical potential accounts for spatial variations in electron density and is of the form mðr; T Þ ¼ mo ðT Þ þ fðrÞ, ! where r f ¼ F. Consequently, for slow variations in electron density, the electrochemical potential mðr; T Þ increases in regions where the density is larger. The phase space description is too important to leave for long, and therefore will be revisited often below. C. Nearly Free Electron Gas 1. The Hydrogen Atom Crystalline solids are aggregates of individual atoms brought together in an orderly arrangement such that, in the case of metals, the outermost electron(s)—originally bound in a Coulomb potential—become free to move about the crystal. The free electron gas model obscures all traces of the bound‐state energy levels and so is unable to, for example, explain the optical spectra of solids or the transition from metallic to semiconducting or insulating behavior. Heuristic models such as the hydrogen atom indicate how such properties result from an arrangement of outermost electrons loosely bound to an orderly array of ionic cores. It is therefore considered in detail.

12

KEVIN L. JENSEN

A quantum mechanical treatment of the hydrogen atom considers the electron wave function c for a rotationally invariant potential as a product of radial RE;l ðrÞ and angular Yl;m ðy; fÞ functions for which m is the spin quantum number (not mass); for present purposes, there is no profit in retaining it, and so m as a quantum number shall henceforth be ignored. ! For the Coulomb potential V ðr Þ ¼ q2 =4peo r Schro¨dinger’s equation for RE;l ðrÞ, where E specifies energy and l angular momentum, is (

)
  
2 1 @ h l ð l þ 1Þ q2 2 @ r    RE;l ðrÞ ¼ EðkÞRE;l ðrÞ: @r r2 2m r2 @r 4peo r

ð25Þ

The natural length scale is the Bohr radius ao ¼ 4peo
h2 =ðmq2 Þ. Let RE;l ðrÞ ¼ GðrÞekr =r so that (  ) @ 2 2 @ 2 þ ðl þ 1  krÞ þ ð1  ðl þ 1Þkao Þ GðrÞ ¼ 0: @r r @r ao r Expressing G as a power series in r such as XN GðrÞ ¼ C rj j¼1 j

ð26Þ

ð27Þ

in Eq. (26) provides a recursion relation for the coefficients Cj Cnl


 2 nkao  1 , ¼ Cnl1 ao ðn þ 1Þn  l ðl þ 1Þ

ð28Þ

where the principal quantum number n ¼ j þ l þ 1 has been introduced. Consider the l ¼ 0 case for convenience: in the limit of large n, Cn  ð2k=nÞCn1 , that is, G(r) has an asymptotic series expansion characteristic of e2kr, which will dominate the factor of e–kr in RE;l ðrÞ unless the series terminates. Therefore, k ¼ 1/nao, implying that the energy E(k) is quantized to the values of En ¼ afs
hc=ð2ao n2 Þ where the fine structure constant afs ¼
h=ðmcao Þ has been introduced. States with larger l are degenerate in energy and such states are termed orbitals; in hydrogen atom parlance, they are called s, p, d, f, and so on. These sharp (discrete) levels have their analogs for heavier ions such that when these ions are brought together with their attendant outermost electrons, the levels evolve into the band structure of solids. Differences in energy between the various l‐orbitals due to spin‐orbit coupling and relativistic effects are not considered here; they break the energy degeneracy and cause the outermost electrons for the heavier atoms of a metallic character to be s states. The first few s radial functions Rn0 ðrÞ

13

ELECTRON EMISSION PHYSICS

1.2

ao

9ao

4ao

1.0 n=1 4p r 2Rn0(r)

dr

0.8 0.6 0.4 n=2

n=3

0.2 0

0

2

4 6 r [Angstroms]

8

FIGURE 2. Probability density for the radial hydrogen atom function for l = 0.

(where n rather than E is used to indicate the principal quantum number in the subscript) are shown in Figure 2. The expectation values of h1=ri1 ¼ n2 ao are also shown, where the n ¼ 1 case corresponds to the Bohr radius. 2. Band Structure and the Kronig–Penney Model For multielectron atoms, the innermost electrons shield the nucleus from the outermost electron, typically an s‐state electron of higher quantum number n for metals. In fact, if both of the outermost s states are filled, as for barium, then each of the s electrons partially shields the nucleus from the other, thus affecting how that atom rests on a surface of other metal atoms, which in turn impacts, for example, the Gyftopoulos–Levine theory for the work function of partially covered surfaces (see Section IV). The interaction between atoms as they are brought together alters the interatomic potentials such that some of the outermost electrons may be free to roam throughout the lattice. The origin of bands and their characteristics is a staple of solid‐state physics texts (Ziman, 1985; Jones and March, 1985; Kittel, 1996; Ibach and Lu¨th, 1996; Que´re´, 1998); herein it suffices to show that bands arise in a one‐dimensional (1D) model with characteristics that extend to three‐dimensional (3D) crystals. Consider an atom (taken to be a metal) relieved of its outermost electron and immersed in distribution of electrons in a uniform background positive charge otherwise known by the descriptive moniker jellium. Electrons are attracted to and therefore cluster about the ionic core, shielding it and screening the Coulomb potential of the core as experienced by other electrons. The change in density dr causes a change in the electrochemical

14

KEVIN L. JENSEN

potential dm ¼ df, and so 1 ð ð dr 2 DðEÞebðEmÞ 3 dfFD ðEðkÞÞ ¼ ¼ b k dE; d 3 2 df ð2pÞ df ð1 þ ebðEmÞ Þ

ð29Þ

0

where the density of states per unit volume of the crystal, defined as the number of states between E(k) and E(k) þ dE, is given by DðE ÞdE ¼ ð2pÞ3 4pk2 dk ¼

m pffiffiffiffiffiffiffiffiffiffi 2mE dE; 2p2
h3

ð30Þ

where the second expression is a consequence of the parabolic relation between E(k) and k (the factor of 2 for spin has not as yet been included). Embedded in the integral are terms that can be rewritten as 1=½ð1 þ ex Þð1 þ ex Þð1=4Þexpðx2 =4Þ, implying that the integrand is sharply peaked about the Fermi level for general temperatures characteristic of electron sources. By comparison, D(E) does not vary appreciably compared to the remainder of the integrand and may be replaced by D(m) and removed from the integral. For bm  1, the remaining integral is unity, so drðrÞ  DðmÞdfðrÞ:

ð31Þ

Therefore, the terms other than D in Eq. (29) tend to conspire and act very much like a Dirac delta function, a feature that becomes uncommonly useful in the following text. For rotationally symmetric potentials, Poisson’s equation is r2 @ r ðr2 @ r dfÞ ¼ 2 ðq =e0 Þdrrecall that f is an energy and r a number density so that the traditional minus (–) sign is absent—and therefore it follows that q2 expðkT F rÞ 4peo r 0 11=2 sffiffiffiffiffiffiffiffi 2 q DðmÞA 4kF ¼@ ¼ eo pao

dfðrÞ ¼ kTF jbm  1

ð32Þ

the Thomas–Fermi screening length is given by 1/kTF. A metallic‐like electron ˚ . For density of 0.1 moles/cm3 implies the screening length is 1/kTF ¼ 0.5854 A pedagogical reasons, however, consider a smaller density associated with a simple lattice of spheres of radius n2ao where n ¼ 3, for which the screening ˚ . A cross‐section of such potentials is shown in Figure 3 length is 1.131 A where the n ¼ 2 and 3 energy levels of the hydrogen atom are shown for comparison (though the energy levels of the potential given in Eq. (32) will be higher).

15

ELECTRON EMISSION PHYSICS

Potential [a.u.]

0

q2 exp(−kTF r)/4peor

a

−1 n=3 −2 −3

n=2

−4 −20 −15 −10

−5 0 5 Position [a.u.]

10

15

20

FIGURE 3. Screened Coulomb potential (red) and multiple adjacent potentials (black).

The derivation of Eq. (32) presumed that bm  0, but this need not be so for semiconductors, where, because the carrier density is orders of magnitude smaller, the chemical potential can be negative. When bm  1, then dfFD /df  bfFD df, in which case kTF becomes  kTF jbm  1 ¼

q2 r eo kB T

1=2 :

ð33Þ

Bare charges are therefore screened by a redistribution of the electron gas surrounding them. If the charge is inside a material with a dielectric constant of Ks, then eo ) Kseo in kTF. The small resistance of metals implies that some fraction of the available electrons are relatively free to move about; such a ‘‘free electron’’ model was developed by W. Pauli and A. Sommerfeld to treat metals, in which a weakly bound valence electron propagates in a lattice of nuclei with their tightly bound core electrons. What, then, is the consequence of these periodic disturbances on the free electrons’ motion? The 1D Kronig–Penney model gives a qualitative sense of what arises (Kronig and Penney, 1931). Consider a square barrier periodic potential V(x) of well width a, barrier width b, such that V(xþaþb) ¼ V(x), and barrier height Vo. According to Bloch’s theorem, the wave function is then given by cðxÞ ¼ uðxÞexpðikxÞ, where k ¼ 2pj=L, L is the (macroscopic) region defining the crystal, and u(x) is a periodic function in x with period (aþb). If barrier regions are designated by cI and the well regions by cII, Schro¨dinger’s equation is

16

KEVIN L. JENSEN

8 < @2

9 =

2  @ 2 2 þ k u ðxÞ ¼ 0 þ 2ik  k  k o :@x2 ; I @x 8 9 < @2 =

 @ 2 2 þ k u ðxÞ ¼ 0 þ 2ik  k :@x2 ; II @x

ð34Þ

hko Þ2 =2m. Solutions are where EðkÞ ¼ ð
hkÞ2 =2m andVo ¼ ð
uI ðxÞ ¼ Aexp½ðkv  ikÞx þ Bexp½ðkv þ ikÞx uII ðxÞ ¼ Cexp½iðk þ kÞx þ Dexp½iðk  kÞx

ð35Þ

where k2v  k2o  k2 . Continuity of the wave function implies that uI ð0Þ ¼ uII ð0Þ and periodicity implies that uI ðaÞ ¼ uII ðbÞ; these two equations, along with two more relating the first derivatives, provide four equations for four unknown coefficients. In matrix notation, 10 1 A 1 1 1 1 B CB B C k ik  ð k þik Þ i ð kþk Þ i ð kk Þ v v B CB C ¼ 0 @ eðkv ikÞa eðkv þikÞa eiðkþkÞb eiðkkÞb A@ C A ðkv ikÞeðkv ikÞa ðkv þikÞeðkv þikÞa ðkþkÞeiðkþkÞb ðkkÞeiðkkÞb D 0

ð36Þ The determinant of the matrix of coefficients must vanish for a solution to exist, which specifies the relation between momentum k and energy via k(E):

 k2v k2 cos½kðaþbÞ ¼ sinðkbÞsinhðkv aÞþcosðkbÞcoshðkv aÞ 2kkv

ð37Þ

For k > ko, then kv ) ijkv|, and the RHS develops an oscillatory nature. The magnitude of the left‐hand side is constrained to be  1, whereas the magnitude of RHS can vary substantially depending on parameters, and for k < ko is generally in excess of unity. Therefore, allowable solutions of k(E) occur only in certain ranges, or bands, the widths of which are determined by how quickly the RHS varies with kva. Consider two limits: first, in the limit ko ) 0, the RHS becomes cos[k(aþb)], indicating that E ¼ ðh
kÞ2 =2m, or

17

ELECTRON EMISSION PHYSICS

1.0 0.9 0.8

k (Energy)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.2

0.4 0.6 k (Barrier)

0.8

1

FIGURE 4. Transition from discrete levels to bands as the barrier k value increases.

10 5 0 −5 −10 1

0 0.5

0.5 0

1

FIGURE 5. Surface plot and contour map based on Eq. (37).

the free electron result, as expected. In the opposite limit, when ko ) 1, solutions exist only when tan ðbkÞ 2k=ko ! 0, or k  jp for integer j, which is the square well limit. For intermediate values of ko, the discrete energy levels of the square well merge into the continuum levels of the free electron, as shown in Figures 4 and 5.

18

KEVIN L. JENSEN

The consequences of the previous treatment indicate that the wave function of electrons above the potential barriers more or less mimics free electron wave functions and that the extent of the band gap is dependent on the magnitude of the potential barrier. That this is not merely an artifact of the square barrier potentials considered is seen by investigating a smooth sinusoidal potential. In bra‐ket notation, consider a 1D region of width L with (unperturbed) basis states defined by hxjni ¼ L1=2 expðikn xÞ ¼ L1=2 expði2pnx=LÞ L=2 ð 1 1X jnihnj 1¼ jxihxjdx ¼ L N n

ð38Þ

L=2

such that the distance between adjacent sites (e.g., atoms) is L/N. Introduce creation and annihilation operators a{ and a such that a{ jni ¼ jn þ 1i _ and ajn i ¼ jn  1i, and a potential operator V ¼ Vl fða{ Þl þ al g so that ^ j0 ¼ 2Vl cosðkl xÞ. We have xjV

 ^ ^ 0 þV^ ðjn0 i þ jn1 iÞ ¼ ðE0 þ E1 Þðjn0 i þ jn1 iÞ; ð39Þ Hjni ¼ E jni ) H where the subscript indicates the order of the approximation for basis states defined by Eq. (38). It follows from the orthogonality relation hnjmi ¼ dnm , where dmn is the Kronecker delta function, that 

E1 ¼ hn0 jV^ jn0 i ¼ Vl hn0 jðn þ lÞ0 þ hn0 jðn  lÞ0 ig ¼ 0: ð40Þ that is, the presence of the perturbation potential does not alter the free electron relation E0 ðnÞ ¼ ð
hkn Þ2 =2m to first order (i.e., there is no first‐order change in energy). However, the density becomes X hxj j ih j jV^ jn0 ihn0 jxi þ c:c: 0 0 jhxjnij2 ¼ jhxjn0 ij2 þ ðnÞ  E0 ð jÞ E 0 j6¼n

ð41Þ X X hxj j 0 ih j 0 jV^ jn0 i n0 jV^ j j0 h j0 jxi 0 0 þ ½E0 ðnÞ  E0 ð jÞ½E0 ðnÞ  E0 ð j 0 Þ j6¼n j 0 ¼ 6 n where c.c. indicates complex conjugate. In the first summation, as a consequence of the creation/annihilation operators comprising V, it follows that only those terms for which j ¼ n 1 survive, and these can be combined to yield D E X hxj j0 i j0 jV^ jn0 hn0 jxi þ c:c: 4Vl cosðkl xÞ ¼ : ð42Þ E E ðnÞ  E ð jÞ ðlÞ  E0 ð2nÞ 0 0 0 j6¼n With a commensurately greater effort, the last double summation can be combined to give

19

ELECTRON EMISSION PHYSICS

E D ED ED ED _ _ X X xj j 00 j 00 jV jn0 n0 jV j j0 j0 jx j6¼n j 0 6¼n

¼ 8 9 ð43Þ < 2 2Vl E0 ðlÞ þ E0 ð2nÞ= cosð2kl xÞ þ E0 ðlÞ  E0 ð2nÞ; E0 ðlÞ½E0 ðlÞ  E0 ð2nÞ :

½E0 ðnÞ  E0 ð jÞ½E0 ðnÞ  E0 ð j 0 Þ

With the introduction of v ¼ 2mVl =
h2 , Eqs. (41)–(43) become 

 
2  4cosðkl xÞ v2 l þ 4n2 cosð2kl xÞ þ 2 jhxjnij ¼ 1 þ v 2 þ 2 2 : ð44Þ l  4n2 l ðl  4n2 Þ l  4n2 2

The integers l and n are generally large, so that jhxjnij2 is generally constant and close to unity except when 2n  l (the pedagogical case of v ¼ 1 and n ¼ 51 is shown in Figure 6). Depending on whether l approaches 2n from below or above, the sign of l – 2n changes from negative to positive, and the density at the ‘‘atomic’’ sites is reduced or increased accordingly. Consequently, a substantially different behavior results for a small change in a parameter characterizing the wave function; it can be shown that to second order, the change in density profile is associated with a change in energy. In other words, a band gap has developed and a forbidden region has occurred for momenta near k(l)  k(2n) as a consequence of the sinusoidal

−27 −8 −1 1 8 27

1.02

|y |2

1.01

1

0.99

0.98 −6

−4

−2

0 k(n)x

2

4

6

FIGURE 6. Eq. (44) for the values of v = 1 and n = 51 for values of l approaching n from above and below.

20

KEVIN L. JENSEN

perturbation—but away from that region, the wave function behaves, to a good approximation, as a free electron (plane wave basis states with energy parabolic in momentum). Near the band gap, of course, the situation is different, but—as shall be seen—emission is generally dominated by momentum states where the ‘‘free electron’’ approximation is good. 3. Semiconductors For intrinsic semiconductors, the Fermi level lies in the band gap between the conduction and valence band levels. Excitations of electrons into the conduction band are accompanied by the creation of ‘‘holes’’ in the valence band. Conditions can be arranged (e.g., by doping) so that a preponderance of electrons or holes occurs. As the distribution of electrons is given by De(E) f(E), the distribution of holes will be given by Dh(E)[1 – f(E)], where the e and h subscripts denote electron and hole, respectively, and f(E) is the distribution in energy of the particles (i.e., the FD distribution). The distinction is required as the ‘‘mass’’ of holes need not equal the electron mass. When charge transport is predominantly carried by electrons, the Fermi level lies closer to the conduction band, and the semiconductor is designated ‘‘n‐type.’’ Conversely, when charge transport is predominantly carried by holes, the Fermi level lies closer to the valence band, and the semiconductor is designated ‘‘p‐type.’’ Moreover, if the Fermi level lies within the band gap and more than 3kBT below the conduction band or above the valence band, the semiconductor is termed nondegenerate. When the Fermi level lies within 3kBT of either band, or falls within either band, the semiconductor is degenerate. Much has been written on the equilibrium carrier concentrations of electrons and holes in doped semiconductors, thereby obviating the need to write more here. For the present, rather, interest lies in the behavior of the semiconductor subject to an applied external field so that carriers migrate to shield out the field in the bulk of the semiconductor. 4. Band Bending Unbound electrons in a material migrate in response to an electric field, thereby shielding the interior of a conductive material from an externally applied electric field. Poisson’s equation relates the unbalanced charge to spatial variations in the potential energy; in one dimension, it is @2 q2 f ð x Þ ¼ ðrðxÞ  ro Þ; @x2 Ks eo

ð45Þ

where the traditional negative sign on the RHS is absent due to r being a number density and f being a potential energy, courtesy of the hidden

21

ELECTRON EMISSION PHYSICS

multiplicative factor of electron charge. Ks ¼ e/eo is the dielectric constant of the material, large for metals and of O(10) for semiconductors. The relationship F(x) ¼ @ xf(x) allows for the substitution @2 f¼ @x2



 @ @ 1 @ 2 f F¼ F ; @x @f 2 @f

ð46Þ

therefore @ 2 2q2 4q2 Nc F ¼ fro þ pffiffiffi @f ks e0 pks eo

ð f

ð1 dy 0

0

dy 1 þ exbð yþmo Þ



pffiffiffi xdx;

ð47Þ

where m(f ¼ 0) ¼ mo and x ¼ bE. Performing the integration over y yields F2 ¼ 

2q2 4q2 Nc fro þ pffiffiffi K s e0 pKs e0 b

ð1 0


 1 þ ebmx pffiffiffi xdx: ln 1 þ ebmo x

ð48Þ

For metals, bmo  1 so that to leading order in f, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    3q2 ro f f f ; 1þ F f 1þ 12mo 12mo l 2mo Ks eo

ð49Þ

where the length parameter l(mo) for the canonical metal (ro ¼ 0.1 moles/cm3) is 58.5 nm. Eq. (49) implies that in the limit f  mo, the potential energy exponentially decays into the bulk with a length factor l. At the surface of a metal, the field F is related to an externally applied (vacuum) field Fvac by F ¼ Fvac /Ks; the largeness of Ks indicates that for metals even under high fields, f remains small, and the potential in the interior remains, to a good approximation, flat (e.g., for Fvac ¼ 10 eV/nm and Ks ¼5000, f < 0.0083 eV). For semiconductors, however, the situation is different by virtue of the relative smallness of Ks and ro: the former is of order O(10), and the latter is of such a magnitude that mo is generally negative. Two limits then exist, depending on whether the electron density is degenerate or nondegenerate as a consequence of band bending. For the more familiar nondegenerate case (bm  1),


 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1=2 2q2 ro bf F¼ e  1  bf  Fo ebf  1  bf; bKs eo

ð50Þ

22

KEVIN L. JENSEN

Field [eV/Å]

100 10−1

Parameters T = 300 K Ks = 12.0

10−2

bmo = −5.5238 ro =1017 cm−3

10−3 10−4

Exact

10−5 10−6

0.01

0.1

1 bf

bm

1

bm

−1

10

100

FIGURE 7. Comparison of Eq. (48) to its asymptotic approximations Eqs. (49) and (50).

whereas for the degenerate case (bm  1)  F ¼ 2Fo

 1=2  bm 1=4 bm=2 2 2 ðbmÞ þ 1 e ; p 15

ð51Þ

where, for T ¼ 300 K, Ks ¼ 12, and ro ¼ 1
1017 cm–3, Fo ¼ 2.7922
10–3 eV/nm. A comparison of Eqs. (50) and (51) with Eq. (48) is shown in Figure 7.

D. The Surface Barrier to Electron Emission The origins of the work function are complex and, indeed, depend very much on surface conditions, material parameters, and many‐body physics. A number of intensive treatments exist in books (Modinos, 1984; Jones and March, 1985; Mo¨nch, 1995), and the periodical literature (aside from articles cited in the following text in context, an excellent recent review may be found in Yamamoto, 2006). Such in‐depth treatments are recommended to compliment the treatment here. 1. Surface Effects and Origins of the Work Function Having shown that to a good approximation, electrons in a conducting material move about in a quasi‐free fashion, and therefore that electron motion is well described by plane‐wave basis states, the origin of the barrier to electron emission at the surface of a material, that is, the ‘‘work function,’’

23

ELECTRON EMISSION PHYSICS

becomes readily explicable. It requires a consideration of how the potential and kinetic energy terms become operators in a basis dictated by particle number (Reichl, 1987; Que´re´, 1998; Feynman, 1972). The Hamiltonian of Schro¨dinger’s equation for many electrons is the sum of several terms: their kinetic energy and the interaction of the electrons among themselves (Hel), their interaction with the background (Vel–B), and finally, the self‐interaction of the background (VB), or H N ¼ HelN þ VelB þ VB 0 N PN ð
hkÞ2 q2 X eajrr j þ HelN ¼ i¼1 2m 4pe0 i<j¼1 jr  r0 j N Ð P

2

q VelB ¼  4pe

0

i¼1 N P

q2

VB ¼ þ 8pe

0

i<j¼1

dr Ð

eaj r ri j

ð52Þ

r r0 rþ ðrÞ

j  ij

0

eajrr j drdr0 jrr0 j rþ ðrÞrþ ðr0 Þ

where the factor ear in the Coulomb potential is inserted to enforce convergence—at the end of the evaluations, the limit of a ! 0 is taken. In the language of creation/annihilation operators, a quantity O as a function of position r, momentum k, and spin s, becomes, in field operator notation, ð   Xð   D E ^ jr s c^{ ðr s Þc^{ ðr s Þ, O r, k ¼ dr1 dr2 r1 s1 jO ^r, k 2 2 1 1 2 2

ð53Þ

s1 s2

where for convenience, O is presumed to be spin independent. The notation becomes burdensome quickly, and it is common to introduce a bra‐ket notation that hides the vector nature and includes spin, that is, jri  jr; si and jki  jk; si. Analogously, interpret integrals over dr to indicate integration over dr andP summation Ð Ð over s, and summations over k to be over k and s, that is, let s dr ) dr, and likewise for momentum (though k is discrete due to finite volume). The following relations hold 1¼

P k

Ð jkihkj ¼ drjrihrj

hrjki ¼ V 1=2 expðik rÞ 0

0

ð54Þ 0

hrjr i ¼ ds;s0 dðr  r Þ; hkjk i ¼ ds;s0 dk;k0 The field operators are represented by X X ^ { ðr ,sÞ ) ^ { ðr ,sÞ ) c hrjkiak ; c hkjria{k , k

k

ð55Þ

24

KEVIN L. JENSEN

and so Eq. (53) becomes O^ ¼

X

hk1 jOjk2 ia{1 a2 ;

ð56Þ

k1 k2

where an indicates an annihilation operator for a number state characterized by momentum kn, and similarly for an{. By way of example, the number operator N becomes X X { ð57Þ k j1jk a ¼ a{ a ; N^ ¼ h ia 1 2 1 2 k k k k k 1 2

where, in the absence of a numerical subscript on k, the numerical subscript on a reverts to the k notation. Likewise, the kinetic energy operator (the first term of Hel) becomes * + ^2 X X
h2 k2 h2 k
{ ^ ð58Þ a{ a : T¼ k k a ¼ a 2 1 1 2 k1 k2 k 2m k k 2m

j

j

The potential terms are more involved, although the self‐interaction of the background is straightforward, as it does not involve the electrons; for a uniform background positive charge, rþ ðrÞ ¼ hN i=V (a consequence of global charge neutrality, as the average electron and background densities must be equal) and so ð ð q2 expðajr1  r2 jÞ hN i2 dr dr ^ ¼ 1 2 VB 8pe jr1  r2 j V2 o 0 1 ð ð1 aR q2 h N i 2 2 @e AdR dr ¼ 4pR ð59Þ 8peo V 2 R 0 ¼

q2 hN i2 4p 8peo V a2

Likewise, the electron‐background contribution is ð q2 h N i X expðaj^ri  rjÞ drhk1 j jk2 ia{1 a2 ¼  ^ VelB k k 1 2 j^ri  rj 4peo V 0 1 q2 hNi X @4pA dk1 ;k2 a{1 a2 ¼ k1 k2 a2 4peo V 0 1 q2 hNi @4pA ^ ¼  a2 N 4peo V

ð60Þ

ELECTRON EMISSION PHYSICS

25

and the electron‐electron contribution (the second term of Hel) is   q2 X expðaj^r1  ^r2 jÞ ¼ k k k k ^ a{1 a{2 a3 a4 : Vee 8pe 1 2 3 4 k1 k2 k3 k4 j^r1  ^r2 j o

j

j

ð61Þ

Eq. (61) hides a rather subtle sleight of hand: wave functions for fermions must be antisymmetric (i.e., the sign must change) when particles are exchanged. Consequently, |k1k2i is not as simple as |k1i|k2i. Rather, |k1k2i must be interpreted as the combination of all |k1i and |k2i arranged such that the antisymmetry is manifest and results in the introduction of the 2
2 Slater determinant jk1 k2 i

ðÞ


 1 jk1 i jk2 i ¼ pffiffiffi det ; jk1 i jk2 i 2

ð62Þ

where det indicates that the determinant of the matrix is to be taken, and the superscript minus (–), which shall be ignored as soon as convenient, indicates ‘‘antisymmetric.’’ The generalization of Eq. (62) to more than two particles should be evident. With Eq. (56), |k1k2i ¼ – |k2k1i, as switching particles is tantamount to switching columns in the matrix, resulting in a sign change in the determinant. It then follows *

expðaj^r1  ^r2 jÞ jk3 k4 k1 k2 j j^r1  ^r2 j

+ ¼ ¼

ð 1 eaR dk1 þk2 ;k3 þk4 dR exp½iðk1  k3 Þ R V R ð63Þ 1 4pdk þk ;k þk 1

2

3

4

V a2 þ jk1  k3 j2

where the V in the denominator is volume, not potential—an unfortunate convergence in notation—and the influence of spin has largely been ignored. Proceeding further requires greater attention to the properties of the creation and annihilation operators. The action of the a0 s on the number‐representation kets is pffiffiffiffi aj jn1 . . .nj . . .n1 i ¼ nj jn1 . . .ðnj  1Þ. . .n1 i pffiffiffiffiffiffiffiffiffiffiffiffi aj jn1 . . .nj . . .n1 i ¼ nj þ 1jn1 . . .ðnj þ 1Þ. . .n1 i

ð64Þ

26

KEVIN L. JENSEN

so that

D E a{j aj ¼ nj  h

ai , aj

ai , a{j

 

i  þ

¼ ¼

Dh



a{i , a{j

i E 

¼0

ð65Þ

  nj þ 1 dnj , 0 þ nj dnj , 1 di, j ¼ di, j

where the last line is a consequence of nj ¼ 0 or 1 for fermions, and [A,B] ¼ AB BA. Thus, Vee for zero momentum transfer (i.e., k1 ¼ k3 ) becomes q2 X 4p { { V^ee ð0Þ ¼  a1 a2 a1 a2 k k 1 2 V a2 8peo n o  q2 X { { { ¼ a a ; a  a a a2 1 1 2 1 2 þ k1 k2 2V eo a2  q2  2 ¼ N^  N^ 2 2V eo a

ð66Þ

Combining VB, Vel‐B, and the zero‐momentum transfer component of Vee gives  q2  2 ^ þ N^2  N^ N  2 N N ð67Þ V^B þ V^elB þ V^ee ð0Þ ¼ h i h i 2V eo a2 n limit, hN^n i ¼ N^ and so hNi2  2hN iN^ þ N^2 ¼

In the thermodynamic

2 N^  N^ ) 0. Moreover, in the same limit, hNi/V remains finite, but hTi becomes increasingly large, so that hNi/V is negligible by comparison. The remaining terms are therefore

P
h2 k2 { 1 X0 ak ak  k1 k2 jV^ee ðk1  k3 Þjk3 k4 a{1 a{2 a3 a4 k k1 k2 k3 k4 2 2m ð68Þ

ðÞ { { P
h2 k 2 { 1 X0 ^ ak ak  ¼ k k k j V ð k  k Þjk k a a a a 1 2 ee 1 3 3 4 1 2 3 4 k1 k2 k3 k4 4 2m

H^ ¼

where the prime on the summation indicates that the no‐momentum transfer term (k1 ¼ k3) has been excluded; the overall negative sign for the potential term is a consequence of the ordering of the annihilation operators, and the superscript minus (–) indicates that hk1k2jVeejk3k4i is replaced by hk1k2| Veejk3k4i – hk1k2jVeejk4k3i, which is antisymmetric to a switch in k3 and k4 and therefore balances the sign change when a3a4 are switched.

27

ELECTRON EMISSION PHYSICS

^ ¼ hn1. . .n1jHjn ^ 1. . .n1i can now be evalThe various components of hHi

2 uated. At zero temperature, the states nj are filled until
hkj =2m > m, beyond which they are empty. Consequently, ha{k ak i functions as the Fermi–Dirac distribution function, the finite temperature extension of which was encountered in Eq. (11) for s ¼ 1, and is the probability that the kth state is occupied, or D E a{k ak / f1 þ exp½bðEðkÞ  mÞg1 : ð69Þ The kinetic energy per unit volume is therefore * +  3 ð 1 X
h2 k 2 { 2 1
h2 k2 L
h2 k5F 2 a y ð m  EðkÞ Þ ; a 4pk dk ¼ ) k 2m k k V V 0 2m 2p 10p2 m ð70Þ where y(m – E(k)) is the Heaviside step function and the factor of 2 in the coefficient of the integral is from a summation over spin. The exchange term, as the second component of Eq. (68) is called, only gives the contribution E 1 DX0 ^ ee ðk1  k2 Þjk1 k2 iðÞ a{ a1 a{ a2 ; hk k j V 1 2 1 2 k1 k2 4V 2 ð ð 2 q yðm  Eðk1 ÞÞyðm  Eðk2 ÞÞ dk1 dk2 ) 6 a2 þ jk1  k2 j2 ð2pÞ eo 

ð71Þ

where commutation of the creation/annihilation operators has been used, and the zero temperature limit has been taken to ease the evaluation of the integrals. As the Fermi–Dirac distribution function changes appreciably only near the Fermi momentum, such an approximation is, in fact, rather reasonable. Recall that the prime on the summation indicates the zero‐momentum transfer component

 has already been removed. Introduce k ¼ k1  k2 and k 0 ¼ 12 k1 þ k2 . The integral over k0 is then    

ð ð  0 1   0 1 ! 0 0 dk y½m  E ðk1 Þy½m  E ðk2 Þ ) dk y kF  k þ k  y kF  k  k 2 2 Ð

ð72Þ

The integral yðr  jx  yjÞdx is the volume of a sphere of radius r offset from the origin by y. Consequently, the RHS of Eq. (72) is interpreted as the volume of intersection of two spheres of radius kF, the origins of which are separated by k  kF. Thus

28

KEVIN L. JENSEN

½

Ð

j

jy½k jk  12 kj

1 0 dk0 y kF  k þ k 2

0

F

Ð arccosðk=2kF Þ 3 ¼ 2pk3F yð2kF  kÞ 0 sin ðxÞdx p ¼ ð4kF þ kÞð2kF  kÞ2 yð2kF  kÞ 12

ð73Þ

which (as it should) reduces to (4p/3)kF3 when the spheres overlap (k ¼ 0). The integration over k is then trivial, and results in (where the limit a ! 0 has been taken) 

ð 2kF

q2 ð2pÞ6 eo

0

4pdk

p q2 k4F ð4kF þ kÞð2kF  kÞ2 ¼  : 12 ð2pÞ4 eo

ð74Þ

To this order, the total energy per unit volume U is the sum of Eqs. (70) and (74), and is
2 k5F h q2 k4F  2 10p m 16p4 eo 3 m ¼ mro  m2 5 p3
h2 ao

U ðr; T ¼ 0Þ ¼

ð75Þ

where, in the second line, ro ¼ r(mo,0) from Eq. (22) and the definition of the Bohr radius ao have been used. In the literature, Eq. (75) is not the most common representation; rather, a dimensionless parameter rs (Table 3) is introduced such that TABLE 3 CORRELATION ENERGY TERMS* rs

eke þ eex þ ecor

1.0 2.0 5.0 10.0 20.0 50.0 100.0

1.174(1) 0.0041(4) 0.1512(1) 0.10675(5) 0.06329(3) 0.02884(1) 0.015321(5)

*Correlation energy values as a function of rs from Ceperley and Alder (1980). Parentheses represent the error bar in the last digit.

ELECTRON EMISSION PHYSICS

29


1 4p hN i  ro ¼ ð r s ao Þ 3 V 3

ð76Þ

in terms of which the energy per unit volume is 80 1 0 12=3 9 1=3 > > < 3p2 9 3= @3A  U0 ðr; T ¼ 0Þ ¼ Ry ro @ A > 10r2s 2p 2rs > ; : 2 0 1 2:2099 0:91633A ¼ Ry ro @ 2  rs rs

ð77Þ

where Ry ¼ 13.6063 eV is the Rydberg energy. The next term, generally called the correlation energy ecor (alternately, the stupidity energy, as sardonically suggested by Feynman, 1972) in the rs expansion is an arduous exercise that is fortunately well treated elsewhere. It accounts for the difference between the total energy and the sum of the kinetic energy and exchange term. An indication of what is entailed can be inferred from the following. In the language of Feynman diagrams, the second term in Eq. (68) can be diagrammatically expressed as k1 k1k2 Vee k3k4

k2 V

⇒ k3

ð78Þ k4

Consequently, the analogous potential interaction term in Eq. (71) generates a diagram of the form (where the line has been compacted to a point ( ) for convenience) k1k2 Vee k2k1

ð79Þ

⇒

Eq. (79) is the lowest‐order Feynman diagram to contribute. The higher‐ order ‘‘polarization’’ diagrams give a contribution DU composed of the higher‐order Feynman diagrams ∆U =

+

+

+…

ð80Þ

where, for sake of convenience, labeling and arrows are suppressed. All such polarization diagrams must be summed to remove the divergence that occurs

30

KEVIN L. JENSEN

for low momentum transfer. A tedious calculation (Feynman, 1972; Que´re´, 1998) shows that including these diagrams results in the small rs expansion 0

DU ðr; T ¼ 0Þjrs !0

1 2 ¼ Ry ro @ 2 ð1  ln ð2ÞÞ ln ðrs Þ  0:096 þ Oðrs ÞA p

ð81Þ

¼ Ry ro ð0:06218 ln ðrs Þ  0:096 þ Oðrs ÞÞ where the term in parentheses is identified as ecor. The terms eex and ecor of Eqs. (77) and (81) represent the low rs , or high electron density, limit of the exchange‐correlation energy term. In the low‐density, or large rs limit, as shown by Wigner, the electron gas ‘‘crystallizes’’ into a lattice. Wigner suggested ecor  0.878Ry /(rs þ 7.79) (Haas and Thomas, 1968), although the form due to Ceperley and Adler (Kiejna and Wojciechowski, 1996)  DU ðr; T ¼ 0Þ  Ry ro

0:862849  pffiffiffiffi rs þ 3:22016 rs þ 3:03546

 ð82Þ

is perhaps better. The various contributions are shown in Figure 8.

0.1 0.05

Energy [Ry]

0

Al Si@ 1E19

−0.05 −0.1 −0.15 −0.2

Corr (rs<<1)

Mg Cu Ag Cs Au Ba Na 1

Corr (wigner) Corr (C&A) Total Metals 10 rs

100

FIGURE 8. Exchange and correlation energy and the position of various metals on the total curve as a function of the (dimensionless) radius parameter rs.

ELECTRON EMISSION PHYSICS

31

The inference from Eqs. (22) and (75) that the energy of the system can be expressed in terms of the density is correct: the ground‐state energy of an interacting electron gas is given as a functional of the density (Hohenberg and Kohn, 1964). Minimization of the energy with respect to the constraint Ð that rðrÞdr ¼ N ¼ constant, analogous to the procedures leading to Eq. (12), serves to relate ro to the effective one‐body potential under which an electron in the material is considered to move (Jones and March, 1985). The variation of Vxc from deep in the bulk of the material to the vacuum outside the surface allows for the determination of the largest component of the work function. The exchange‐correlation potential is determined by the functional derivative of the exchange‐correlation energy Exc (i.e., U0 þ DU without the kinetic energy term), or Vxc ðrÞ ¼

dExc ½rðrÞ : drðrÞ

ð83Þ

Technically, Eq. (83) is valid for a uniform electron density—its application to a non‐uniform density makes use of the local‐density approximation (LDA) in which Vxc is calculated for a small‐volume element for which the local density is ro. Surprisingly, however, the procedure continues to work well even when the electron density is rapidly varying, as near an ionic core—but more to the point here, near the surface of a metal—and therefore, the LDA is made in almost all density functional calculations (Sutton, 1993). Consider the case of sodium, for which rs ¼ 3.93 in bulk (Kittel, 1996). Therefore, Vxc(rs ¼ 3.93) – Vxc(rs ¼ 1) is 5.266 eV and m ¼ 3.245 eV. Their difference corresponds to a potential change from bulk to vacuum of 2.021 eV (Ceperley and Adler) or 2.073 eV (Wigner), values surprisingly close to the work function of sodium (F ¼ 2.29 eV; Haas and Thomas, 1968). The success of sodium is quickly tempered by the divergence of the method for other metals, inviting the justifiable suspicion that the physics of other effects is being neglected. These effects are discussed next. 2. Ion Core Effects As alluded to in the discussion of the Kronig–Penney model, the ion cores associated with the metal atoms cannot be neglected. Crudely, the ionic cores can be thought of as residing in spheres of radius rs ao surrounded by an electron charge cloud, so that the overall sphere is neutral and the spheres in the crystal, if nonoverlapping, therefore noninteracting (except for, perhaps, van der Waals and repulsive forces; Herring and Nichols, 1949—effects that are small for metals and therefore judiciously ignored). Inside the

32

KEVIN L. JENSEN

rs sphere, the electron density is relatively constant except near the core. Two contributions to the energy exist: the electron cloud interacting with itself, and with the ion core. Within the spherical approximation, the electron self‐ and core interactions are easily determined: the cloud–ion core interaction is given by  ð rs ao  3Ry nc q eei ¼  nc ; ð84Þ qro 4pr2 dr ¼  4pe r rs 0 0 where the term in parentheses is the Coulomb potential of a ion core with charge ncq. Similarly, the cloud self‐interaction term is "   # ð rs ao 3Ry 2 ns q r 3 ns q eee ¼ n; ð85Þ ro 4pr2 dr ¼ 4pe r r a 5rs s 0 s o 0 where ns is the number of electrons in the sphere (henceforth, for ease of discussion, ns and nc, will be taken as unity—that is, ignored—to avoid the discussion becoming needlessly complex), and the terms in the integrand are the Coulomb potential for a charge nsq, the charge within a uniform charge density sphere of radius r (note that r is a length, but rs is dimensionless) and the charge in the shell comprised of the product of the charge density with the differential volume element of the shell. Consequently, a  term ei ¼  9Ry =5rs may be added to the exchange‐correlation energy to account for the effect of the ion cores of the metal. The inclusion of ei contains a sleight of hand: the ion core is not a bare charge, but rather is surrounded by an inner cloud of electrons that shield it from the valence electrons. Therefore, the ability of the valence electron to penetrate the core is circumscribed. A simple approximation is to assume that up to a radius ai (a notation evocative of the Bohr radius ao), the core region excludes the outer electron completely, but for r > ai  ri ao , the potential of the core is a simple Coulomb potential. While such a ‘‘pseudopotential’’ approximation appears to be draconian, in fact it performs rather well (Ashcroft, 1966; Que´re´, 1998). Eqs. (84) and (85) must therefore be modified to exclude the core region, both in the e terms, but also the density ro term, giving 0 1 ð rs 3  r3 6Ry r i  r@ 3 ei )  3  1Adr rs  r3i rs  r3i ri 0

¼

3Ry @3r5s 5

 5r2i r3s þ

3 2 rs  r3i

1

2r5i A

ð86Þ

ELECTRON EMISSION PHYSICS

33

The ground state of the system may be approximated by the following argument (Jones and March, 1985). Within the rs sphere, the electron has a wave function of the form ck ðrÞ ¼ eikr ½ j0 ðkrÞ þ Kan0 ðkrÞ, where j0(x) and n0(x) are spherical Bessel functions of order zero, and a is a scattering length from the zero‐momentum limit of the scattering amplitude f(y), but which for simplicity will be evaluated using the Born approximation, for which  2  ð ai 2m sinðkrÞ q a2 ð87Þ a ¼  2 lim r2 dr ¼  i ; 4peo r a0
h k!0 0 kr where ai ¼ riao. The factor of K is found by requiring that the first derivative of the wave function vanish at rs, giving K 2 ¼ 3a=ðrs ao Þ3 . A constant energy term eo ¼

ð
hK Þ2 r2 ¼ 3Ry i3 2m rs

ð88Þ

is then added to the overall energy expression. The total energy is the sum of the kinetic, exchange‐correlation, ion core, and ground‐state energies, or E ¼ Uo þ DU þ Ei þ Eo. With the removal of the kinetic energy term in Uo, the variation of the remainder with respect to the electron density r(rs), as per Eq. (83), gives rise to the largest component of the potential barrier that an electron experiences at the surface of a material. 3. Dipole Effects Due to Surface Barriers To the potential resulting from exchange‐correlation energy and ion core terms must be added any effects due to self‐consistency. At the surface, the quantum mechanical, or wave, nature of the electron allows the electron to be found in the classically forbidden region of the surface barrier, which serves to prevent electron escape into the vacuum. A model for the estimation of the magnitude of the dipole effect is obtained by considering Schro¨dinger’s equation for a potential barrier in the form of a wall of height Vo and width L, or V ðxÞ ¼ Vo ðxÞðL  xÞ, where  is the Heaviside step function (Jensen, 2003a). Wave functions approaching the barrier (from the left) and leaving (to the right) have the form  1

ck ðx < 0Þ ¼ pffiffiffi eikx þ rðkÞeikx 2 c k ðx  L Þ ¼

ð89Þ

tðkÞeikx

where
hk is the momentum of the electron with corresponding energy E ¼
h2 k2 =2m. Momentum‐like variables prove convenient, so introduce

34

KEVIN L. JENSEN

1=2 ko  2mVo =
h2 k2  k2o  k2

ð90Þ

The requirement to match the wave function and its first derivative at each (abrupt) change in potential is concisely expressed in the matrix equation for k < ko (for k > ko, k ) ik) 

1 1 ik ik



1 rðkÞ



 ¼

1 k

1 k



ekL kekL

ekL kekL

1 

eikL ikeikL

eikL ikeikL



tðkÞ 0



ð91Þ The solution of Eq. (91) for r(k) and t(k) is straightforward albeit monotonous. It suffices to quote the results, the general methodology of the calculation being deferred to the development of the emission equations. The solutions are

tðkÞ ¼

2kkeikL 2kkcoshðLkÞ þ iðk2  k2 ÞsinhðLkÞ

iðk2 þ k2 ÞsinhðLkÞ rðkÞ ¼  2kkcoshðLkÞ þ iðk2  k2 ÞsinhðLkÞ

ð92Þ

in the limit kL  1, then jrðk < ko Þj  1, indicating that for electrons with energies below the barrier maximum, total reflection occurs when the barrier is tall, wide, or both, and a reflection (assuming such a pun is permissible) of the exponential decay of the electron density within the classically forbidden region under the barrier. To find ck(x) to the left of the barrier, let r be given as rðkÞ  RðkÞexpð2i’ðkÞÞ 8 2 391 < = 2kk 5 RðkÞ ¼ 1 þ 4 2 : ko sinhðLkÞ ; tanð2’ðkÞÞ ¼ 

2kk cosh ðLkÞ ðk2  k2 Þ sinh ðLkÞ

ð93Þ

ELECTRON EMISSION PHYSICS

35

For tall, wide (or both) barriers, the reflection coefficient is, to a good approximation, independent of barrier width, a consequence of cosh(z)  sinh(z) for large z so that jck ðxÞj2 ¼

 1

1 þ R2  R cos½2ðkx þ ’ðkÞÞ 2

ð94Þ

 1  cos½2ðkx þ ’ðkÞÞ When k2o  k2 in particular, Eq. (93) indicates that ’ðkÞ   k=ko  kxo , so that to leading order the effect of a tall barrier, regardless of width, is to simply shift the density to the left by an amount inversely proportional 1=2 to Vo . The impact of the shift by xo is most readily seen in the zero‐temperature limit for density, which (as for all things quantum mechanical)P is altered from ^ ¼ i ni jiihij in the Eq. (14): the density matrix (Shankar, 1980) is defined as r number representation, where ni ¼ 0 or 1 for fermions. In the momentum representation, then ð X 2 ^¼ fFD ðEðkÞÞjkihkjdk r n jkihkj ) ð95Þ k k ð2pÞ3 which contains some more sleight of hand: the k‐kets to the left contain three momentum components and spin, as in Eq. (54), whereas to the right, the spin states have been summed over, the vector nature of k is made explicit, and the transition to the continuum limit has been made, where nk is replaced by the FD distribution function. Consequently, the k of Eq. (94) is but one momentum component of the k in Eq. (95), say kx. The density of interest here is the diagonal of the density matrix, or hxj^ rjxi  rðxÞ, which becomes ð1 ð1 2 rðxÞ ¼ jck ðxÞj2 dk 2pk⊥ dk⊥ f FD ðEðkÞÞ; ð2pÞ3 1 0 ð ð96Þ 1 1  f ðkÞjck ðxÞj2 dk 2p 1 where kx ) k, and the 1D ‘‘supply function’’ f(k) has been introduced. f(k) is seen to be the FD with the transverse momentum components integrated over; its sly introduction via Eq. (96) belies its significance, as its usage in electron emission theory is ubiquitous. With the assumption of a parabolic relationship between energy and momentum, it is straightforward to show that f ðkÞ ¼

m lnf1 þ exp½bðm  EðkÞg; pb
h2

ð97Þ

36

KEVIN L. JENSEN

where
hk is understood in Eq. (97) to be the 1D momentum into (or away from) the

barrier.  In the zero‐temperature limit b ) 1 so that f ðkÞ ) ð2pÞ1 k2F  k2 ðkF  kÞ, for which r(x) becomes ð kF

2  2 kF  k2 ½1  cosð2kðx  xo ÞÞdk lim rðxÞ ¼ 2 b!1 ð2pÞ 0 8 9 ð98Þ k3F < cosðzÞ sinðzÞ= ¼ 2 1þ3 2 3 3 3p : z z ; where z ¼ 2kF ðx  xo Þ and the coefficient, equal to ro, is familiar from Eq. (22). The behavior of r(x) is shown in Figure 9, along with a hyperbolic tangent fit ra(x) and a step‐function ri ðxÞ ¼ ro ðxi  xÞ, where xi is the location of the origin of the background positive charge. The oscillations visible, due to the trigonometric functions in Eq. (98), are known as Friedel oscillations and are a consequence of the wave nature of the electron. The value of xi is found by demanding global charge neutrality, or ð1 ½re ðxÞ  ri ðxÞdx ¼ 0 ð99Þ 1

A bit of manipulation shows that Eq. (99) is equivalent to (ð ( ) ) 1 3 1 i lim Re þ

 eis ds þ ðxi  xo Þ ¼ 0 2kF d!0 s 2 þ d2 s s 2 þ d2 0

ð100Þ

1.2 1.0

r(x)/ro

0.8 Excess (+) charge Excess (−) charge

0.6 0.4 0.2 0 −6

r(x)/ro ra(x)/ro 1.244 ra(x)/ro 2.554 Ion −4

−2

0

2kF(x–xo)/p FIGURE 9. Electron density compared to the bulk value and the nature of Friedel Oscillations at the surface.

ELECTRON EMISSION PHYSICS

37

The first term contains integrals familiar from the calculus of residues and is straightforwardly shown to be ð1 ( Re 0

)  1 i p p

þ

 eis ds ¼ ed  2 1  ed : 2 2 2d s2 þ d s s2 þ d 2d

ð101Þ

It follows xi ¼ xo 

3p 8kF

ð102Þ

˚ and For example, if ro ¼ 0.1 moles/cm3 and F ¼ 4.5 eV, then xo ¼ 0.61415 A ˚ xi ¼ –0.35749 A. Evaluating the dipole resulting from the charge distribution in Figure 9 requires a bit more finesse. Integration over the field F ¼ @ xf gives the dipole contribution Df: Ð1

1

ð@ x fÞdx ¼ 

Ð1

1

q2 ¼ eo

 x @ 2x f dx

ð1 1

ðx  xo Þðre ðxÞ  ri ðxÞÞdx

ð103Þ

where integration by parts, the vanishing of the field at the boundaries, Poisson’s equation, and global charge neutrality have been used. Let s ¼ 2kF (x – xo) and D ¼ 2kF (xi – xo) ¼ –3p/4. The integrand for the electrons is then proportional to sfcosðsÞ=s2  sinðsÞ=s3 g ¼ @ s ½sinðsÞ=s so 8 9 0 1 ð0 ð = q2 ro < 0 d @sinðsÞA ds  Df ¼ 3 sds ; s 4eo k2F : 1 ds D  QkF

32  3p2 ¼ 16p

ð104Þ

where the definitions of Q and ro have been used. For the canonical example ˚ 1, Df ¼ 0.4153 eV. of a 0.1 moles/cm3 density for which kF ¼ 1.213 A Two of the approximations undermine the clean simplicity of Eq. (104): the approximation ’(k)  kxo has been used, and thermal effects (which were neglected) come into play. The former causes the Friedel oscillations to be more pronounced than would be evident from a numerical evaluation of Eq. (96). The latter introduces electrons with energy greater than the Fermi level that penetrate more deeply into the barrier, causing the dipole term to be larger. The effects also trickle down to modify the definition of xi. A more phenomenological theory (Jones and March, 1985; Smith, 1969)

38

KEVIN L. JENSEN

is to consider the approximate electron density profile to be defined by ra ðxÞ ¼

ro : 1 þ exp½2lkF ðx  xi Þ

ð105Þ

If l is chosen such that @ xre(x) ¼ @ xra(x) at x ¼ xi along with Eq. (98) and D ¼ –3p/4, then l¼

 36 12

cosðDÞ þ 4 D2  3 sinðDÞ ¼ 1:24356: 3 D D

ð106Þ

The integral for Df is then trivially evaluated by Dfa ¼

q2 r o 2e0 ðlkF Þ2

ð1 0

s 2p ds ¼ 2 QkF : 1 þ es 9l

ð107Þ

Use of Eq. (106) makes Eq. (107) substantially larger than Eq. (104) (the 0 value l0 that would give equality between Df and Dfa is l ¼ ð4p=3Þ 1=2 ½32  3p2  ¼ 2.5546). The discrepancy lies with the fact that the fields generated by the charge further away from the interface contribute in the tanh model, whereas for the Friedel‐model, the fields generated by the charge between (2n þ 1)p  2kF(x – xo)  (2n  1)p (i.e., zeros of sin(2kF(x – xo)) vanish, and no net charge exists past x > xo, a relic of the infinite barrier approximation). How much do small excesses of charge inconveniently located elsewhere change the presumed potential profile? In fact, a great deal. For the generic metal example of re ¼ 0.1 moles/cm3, a thin sheet of ˚, charge of width Dx produces a field of re Dx=2eo ; in other words, if Dx ¼ 0.2 A ˚ ngstrom‐scale then a field of 0.11 eV/nm results, indicating that over A distances, electron volt–scale potential differences are created. The ‘‘wings’’ that accompany the tanh model can be clipped by setting the integration limits in Eq. (103) to xi xo. With re replaced by ri, Eq. (107) becomes 2p QkF ½1  M ð2lkF xo Þ 9l2 1 12 X ð1 þ kzÞ kz M ðzÞ ¼ 2 ð1Þkþ1 e p k¼1 k2 Dfa )

ð108Þ

For example, M(2)  0.468, showing that the hyperbolic tangent approximation by itself gives estimates to Df that are a bit too large (see Figure 14). Nevertheless, insofar as Eq. (108) is a reasonable model if a good choice of l is found, it demonstrates that the magnitude of the dipole potential

39

ELECTRON EMISSION PHYSICS

Core (2.71) Core (1.24) Pauling

Core radius [angstroms]

3

A&L

2

1

Cs Rb K Ba Na Li Zr Ag Mg Au Co Cu Pb Ni Zn Al Fe W Element FIGURE 10. The core radius ri compared to the Pauling radius and radii from Ashcroft and Langreth for two values of l (2.71 and 1.24).

contribution is dependent on both the electron density and the height of the barrier, and while crude, it provides a qualitatively satisfactory account of the variation of the dipole term with electron density for simple metals. An estimate of the work function is then obtained from the relation, suggested by Eqs. (83) and (104) (the signs reflect that the vacuum level is at 0) @ ð109Þ ðm þ FÞ ¼  ½rðeex þ ecor  eo þ ei Þ  Df @r Performing the derivative is cumbersome, but the resulting equation yields a rough estimate of F if rc is known. Given the qualitative treatment of Df and the ion core terms here, however, ri is not known and is therefore only qualitatively similar to the actual ionic core radii. Approaches that pay better attention to the ionic core terms, such as by Ashcroft and Langreth (1967), or to the dipole terms, such as by Lang and Kohn (1970), fare better. Nevertheless, if actual work function values are used, then Eq. (109) can be used to predict ri, and the values of ri so found compared to, for example, the Pauling radii. The results of that analysis are shown in Figure 10 for two values of l. Overall, given the approximations, the relationship between ri and the Pauling radius is good, particularly for ‘‘good’’ metals such as cesium, barium, sodium, and potassium (columns 1 and 2 on the periodic table)—but not in all cases, especially those where the impact of d‐shell electron contributions are nontrivial, such as tungsten, iron, and others. Moreover, the qualitative behavior of the core radius in the simple model here correlates well with trends of, for example, Ashcroft and Langreth (1967) (also shown in Figure 10).

40

KEVIN L. JENSEN

+

fq

+

+

L −

−

FIGURE 11. Smoluchowski model of corrugation of a real surface, showing a migration of electron charge and the subsequent creation of a dipole. The “net charge” at the apex is fq.

The 1D model does not account for expressly 3D complications associated with real surfaces; the representation of a surface by a uniform and featureless plane is a substantial idealization. Real crystal surfaces appear corrugated and complicate the work function. Smoluchowski (1941) proposed a surface of regular pyramids and pyramidal depressions mimicking a real surface such that the apexes and valleys become equally and oppositely charged (Figure 11). The base of the pyramid is of length scale L and the charge at the apex of the period is fq (where f is presumed to be 1) over an area L2, equivalent to a surface charge density of fq/L2. Crudely, the potential energy drop across the dipole formed by the pyramids, on average, is then df  D( fq2/L2)/eo ¼ 18.095 eV‐nm f D/L2, where D is the approximate separation of the dipole layers after the charge has partially smoothed out. ˚ , L ¼ 24 A ˚ , and f ¼ 0.3, Considering the pedagogical numbers of D ¼ 2 A then df  0.189 eV. Since different crystal orientations have different corrugated shapes at the surface, the model makes plausible the small differences in work functions that occur for different faces, albeit that it is defective as an actual description of surface atomic structure. A similar argument may be used to explain the tendency of adsorbates that charge either positively or negatively to lower or raise the work function accordingly, such as barium and oxygen on tungsten in a B‐type dispenser cathode, which has a work function several tenths of an electron volt lower than bulk barium (this topic is discussed in the section on the Gyftopoulos–Levine theory). Contours of electron density for surfaces such as tungsten, with or without a barium overlayer, hint at such a triangular structure (see, for example, Figure 3 of Hemstreet, Chubb, and Pickett, 1989). Other effects not treated here also leave their mark (e.g., effective mass differences, field enhancement). Nevertheless, Eq. (109) accounts for the dominant influences and is adequate to anticipate the behavior of the image charge potential near the surface. E. The Image Charge Approximation 1. Classical Treatment Classically, a charged particle outside a conductor causes a redistribution of charge on the surface of the conductor that serves to screen out the external field. For a charge –q a distance x outside a conducting surface, it is a common

41

ELECTRON EMISSION PHYSICS

result, familiar from electrostatics, that to ensure that the surface remains at zero potential, an image charge þq lies at –x from the surface inside of the conductor (Eyges, 1972). The potential energy of the external charge is obtained from the integral over the field between the charged particle and its image, or ! ð1 q2 q2 Vimage ðxÞ ¼ ð110Þ  dx ¼  2 16peo x 4peo ð2xÞ x or Vimage(x) ¼ –Q/x. For a metal surface subject to an external field, and using the bottom of the conduction band as the reference level in energy, the image charge potential is then given by Vimage ðxÞ ¼ m þ F  Fx 

Q x

ð111Þ

where F is the product of electron charge and electric field. Far from the surface, Vimage must be correct at macroscopic distances—if one is flexible about the definition of F—and a more subtle analysis using the exchange‐correlation potential and the jellium model is required to reveal the 1/x dependence (Lang and Kohn, 1973). Near the surface, Eq. (111) is unphysical, plummeting to –1 at x ¼ 0þ. The hyperbolic tangent approximation to the density in Eq. (105) with the exchange‐correlation analysis does serve to qualitatively validate Eq. (111) near the surface. For the generic metal with re ¼ 0.1 moles/cm3, the exchange‐correlation potential associated with the hyperbolic density plus the dipole is compared to the image charge potential in Figure 12. The agreement between the exchange‐correlation potential and the image charge potential is not optimal—a consequence of the approximations behind the hyperbolic‐ tangent density approximation—but the comparison clearly does show that image‐charge–like variations in the potential exist near the surface simply as a consequence of variations in density and the exchange‐correlation potential prescription of Eq. (109). The more careful the theoretical analysis, the better the agreement, further supporting the utility of the image charge potential model of the surface (Kiejna, 1991, 1993, 1999). The shifting of the density by an amount xo as a consequence of Eqs. (94) and (98) creates an expectation that the position x in Eq. (111) should be replaced by x þ xo, so that Vimage(x) ¼ m þ F – F(x þ xo) – Q/(x þ xo). Some (e.g., Lang and Kohn. 1973) give xo with a negative sign—the choice reflects convention—and in their parlance, xo represents a ‘‘center of mass’’ given by Ð1 xr ðxÞdx ð112Þ xo ¼ Ð0 1 e 0 re ðxÞdx

42

KEVIN L. JENSEN

1.2

10

0.8

8

Friedel Tanh model

6

0.6

Tanh + dipole Image potential

0.4

4

Potential [eV]

Density/bulk density

1.0

2

0.2 0

0 F = 0.4 eV/Å

−8

−4

0 Position [Å]

4

8

FIGURE 12. Density (thin red line) showing Friedel oscillations and the related exchangecorrelation potential (thick red line). Also shown are the hyperbolic tangent approximation to the density (thin blue dashed line), and the image charge potential (green dashed line).

The shifting of the potential by an amount inversely proportional to the square root of the barrier height, as well as variations in potential arising from fluctuations in density, are expectations borne out in more rigorous treatments. It is important, however, to bear in mind that the model considered here is idealized: the jellium charge distribution is slightly different than a realistic charge distribution of actual metals, which serves to complicate the specification of the image plane position (see, for example, Forbes, 1998). 2. Quantum Mechanical Treatment A Green’s function approach to the determination of the metal‐vacuum interface potential accounts for potential variation near the surface due to quantum effects and correctly accounts for the asymptotic image charge behavior (Il’chenko and Kryuchenko, 1995; Qian and Sahni, 2002). When the interface is subject to an external field, a Thomas–Fermi approximation (TFA) can quite elegantly provide useful approximations to xo in the image charge potential; accounting for the quantum mechanical nature of the conduction band electrons via the random‐phase approximation (RPA) illuminates the nature of the Friedel oscillations under more general conditions than considered above. The TFA and RPA approaches differ in their approximations to the dielectric function in a metal (Il’chenko and Goraychuk, 2001), and therefore in their predictions, for example, in barrier height, but both validate the qualitative

ELECTRON EMISSION PHYSICS

43

behavior of the classical image charge potential and modifications to it. RPA is preferred as it can, for example, deal with complications related to the shape of the Fermi surface such as occur for real metals as a consequence of the behavior of the dielectric function. The TFA, however, is easier to express concisely and analytically; therefore it is of greater utility in illuminating modifications to the image charge potential. Approximate expressions to V(x) based on the TFA are then given by (Il’chenko and Kryuchenko, 1995) 8 F kx Q > >  e  e2kTF x ð x < 0Þ > > > k x  2x TF o < 0 1 ð113Þ V ðxÞ  > 4 > > 2kTF Q  F @x þ xo A  Q ð x  0 Þ > > : 3 x þ xo where kTF is from Eq. (32), and xo has been redefined as xo ¼ 3/(4kTF). For semiconductors, the image charge term Q is modified for a material with dielectric constant Ks according to   Ks  1 afs
hc ð114Þ Q) Ks þ 1 4 The work function therefore becomes F ¼ 2kTFQ – m. In this sense, xo does scale as 1/(barrier height)1/2, as expected in the discussion surrounding Eq. (112), and the intuition that x should be (more or less) replaced by x þ xo in the classical image charge potential is vindicated. The behavior ˚ and of Eq. (113) is shown in Figure 13 for an applied field of F ¼ 0.4 eV/A –1 ˚ kF ¼ 1.3 A . Appealing though it is, the barrier predicted by Eq. (113) is smaller than that predicted using RPA, and the TFA approach lacks the conspicuous Friedel oscillations in Eq. (98) and Figure 12 that the RPA approach replicates; however, it provides a justification for the development of an analytical image charge potential by showing that quantum and physical effects can be accommodated (however crudely) through modifications to the work function, field, and image charge terms. 3. An ‘‘Analytical’’ Image Charge Potential The analytic image charge potential is a simplistic approximation compared with the RPA approach or treatments where the wave function and the exchange‐correlation and dipole terms are evaluated self‐consistently (Figure 14). The justifications for considering an ‘‘analytical’’ image charge model are as follows: (1) the development of emission equations using approximation formulas for the potential are the most expedient, (2) Friedel

44

KEVIN L. JENSEN

Potential [eV]

10 8 6 4 TFA Image

2 0

0 5 10 Position [angstroms]

15

FIGURE 13. Comparison of the Thomas–Fermi approximation [Eq. (113)] to the image charge approximation.

1.5 tanh approx. Friedel Approx

∆f[eV]

1.0

0.5

0.0 0.001

0.01

0.1

Density [1024 #/cm3] FIGURE 14. Comparison of the dipole term calculated from the hyperbolic tangent approximation compared to the density using the Friedel approximation.

oscillations and variation in the bulk are secondary in the development of the emission equations; and (3) the impact of both field and temperature are easily accommodated. The effects to be included are dealing with the consequences of the origin of the background positive charge in the jellium model not being coincident with the origin of the electron distribution and the temperature of and field dependence of barrier height. The shift in the x parameter by xo has been treated previously. Regarding the temperature and field dependence of the barrier height, the electron gas temperature and applied field have an impact on the magnitude of the work function for analogous reasons. Both allow greater penetration of the emission barrier

45

ELECTRON EMISSION PHYSICS

by either a larger population of more energetic electrons (for temperature) or the ability of the electrons to penetrate (for field). Coupled with the dependence of the exchange‐correlation potential on density, as well as the behavior of the dipole contribution, the height of the barrier above the Fermi level changes. It is therefore expected that terms accounting for changes in the work function that depend on temperature, field, and extent of the dipole will need to be introduced, presumably resulting in a change in the definition of F: the image charge potential should therefore resemble V ðxÞ ) mðT Þ þ fðF ; T Þ  Fx 

Q ; x þ xo

ð115Þ

where the use of f accommodates field‐ and temperature‐dependent effects. The origin of the background positive charge and the electron gas have been treated as equivalent, but the shifting of the density in response to the barrier height shows that the assumption is not strictly valid. Let the separation between the electron gas origin and the background positive charge origin be dxi. The density of the background positive charge is ro, and, due to global charge neutrality, only the potential drop Dfi across the slab needs to be added to the work function estimate. It follows from Poisson’s equation that  ð ð x0 ro dxi 8 00 Qk3F dx2i : Dfi ¼ dx dx0 ¼ ð116Þ 3p e0 0 0 While dxi can be found for simple models, in general it must be found by enforcing global charge neutrality in a numerical simulation (an example of which is found in Jensen [1999, 2000], although the present treatment has changed) as it depends on electron temperature, applied field at the surface, and the barrier height. Assuming that has been done, what remains of the image charge approximation? A rearrangement of terms results in Vanalytic ðxÞ ¼ mðT Þ þ Feff  F ðx þ xo Þ 

Q , x þ xo

ð117Þ

where an ‘‘effective’’ work function Feff is introduced. Eq. (117) does not appear to confer any benefit, as it appears to substitute one set of difficult parameters for another, but its advantage lies in the fact that in estimates of emission current, the classical image charge figures prominently in analytical formulae, particularly the Richardson–Laue–Dushman (RLD) equation for thermionic emission and the Fowler–Nordheim (FN) equation for field emission. The introduction of an effective work function allows for subtle effects related to temperature and field on the barrier height and the transmission probability to be ‘‘smuggled’’ into the RLD and FN formulas without necessitating grueling numerical or analytical effort provided the behavior of Feff can be ascertained. What might that behavior be like? Experimentally,

46

KEVIN L. JENSEN

the work‐function variation with temperature is linear, with the coefficient of the temperature material dependent (see (Haas and Thomas, 1968) for a listing of coefficientspffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for various elements). Similarly, from the expression xo ¼ k1 ¼ h
= 2m ðm þ fÞ, where f is the height of the barrier above the o Fermi level, the lowering of the image charge barrier by the application of field (known as Schottky barrier lowering) will relate to the shifting of the density profile, and therefore to the value of dxi. Numerical findings suggest that the relation between the barrier height and Fxo is also linear to a good approximation. Therefore, the effective work function should resemble Feff ðT; F Þ  Fo þ ao kB T þ

8 Qk3 dx2 þ 2Fxo ; 3p F i

ð118Þ

where parameters such as the (dimensionless) parameter ao are from the thermal dependence of the work function, known from the literature (see examples in Table 3) and the others chosen so that Feff is equal to the experimental work function at zero field and a known temperature. The temperature and field variation of dxi is slight, and as a pragmatic matter, it is adequate to absorb the term in which it resides into Fo, or to parameterize it based on numerical simulations. Regardless of the method used to determine the temperature and/or field dependence of the work function, the determination of emission current from the classical image charge potential is convenient as long as the fact that the work function in it depends on temperature and field in the manner suggested by the discussion surrounding Eq. (103) is kept in mind. With that caveat, any equation in the following text dependent on the image charge potential can be trivially modified by the usage of Eq. (118) for the replacement of the work function with an ‘‘effective’’ work function (see Table 4). TABLE 4 PARAMETERS OF WORK FUNCTION* Atomic number

Element

Fo

ao

47 79 56 72 25 42 41 14 74

Ag Au Ba Hf Mn Mo Nb(100) Si W

4.31 4.25 2.3 3.6 3.83 4.33 3.95 3.59 4.52

0.12 0.17 5.80 1.62 1.28 0.12 0.35 2.67 0.70

*F(T) ¼ Fo þ ao kBT; adapted from Haas and Thomas, 1968.

47

ELECTRON EMISSION PHYSICS

II. THERMAL AND FIELD EMISSION A. Current Density In the discussion of the hydrogen atom, the Kronig–Penney model, and the dipole contribution to the work function, Schro¨dinger’s equation was ‘‘smuggled’’ into the discussion; when analyzing the surface barrier, a methodology was introduced by fiat to yield transmission and reflection coefficients in Eqs. (89) and (92), which are needed to determine the extent to which the wave function penetrated the barrier. A return to the central issue in the development of the emission equations, namely, the barrier problem— but now with the proper formalism—is desirable. In contrast with perhaps standard treatments, the approach here is to start with a distribution function and move to the Schro¨dinger representation to allow for the introduction of the classical approach followed by its quantum extension. 1. Current Density in the Classical Distribution Function Approach The Boltzmann equation in Eq. (24) is familiar from the classical statistical mechanics of an ideal gas (Reichl, 1987; Reif, 1965), for which the number of particles dn ¼ f(x,k,t) dx dk in a small region at time t is conserved. After a time t0 ¼ tþdt, all the particles must be accounted for, or f ðx; k; tÞdxdk ¼ f ðx0 ; k0 ; t0 Þdx0 dk0 :

ð119Þ

hk0 ¼
hk þ ðF =mÞdt, so that To order O(dt), x0 ¼ x þ ð
hk=mÞdt and
the Jacobian is (the 1D case is shown, but the 3D case represents a straightforward generalization)

]x x0

]k x0



]x k0



1 ¼ hdt ]k k0





0

¼ 1: 1


ð120Þ

Therefore dx0 dt0 ¼ dxdt, from which the conclusion f ðx; k; tÞ ¼ f ðx0 ; k0 ; t0 Þ follows. A Taylor expansion then shows 8 <]

9 f ðx þ dx; k þ dk; t þ dtÞ
f ðx; k; tÞ
hk ] F ] = ¼ þ þ f ðx; k; tÞ; 0¼ :]t m ]x
h ]k; dt ð121Þ where F ¼ ]x V ðxÞ (compare Eq. (24)). ‘‘Moments’’ of a general distribution function are defined by

48

KEVIN L. JENSEN

ð hOðr; kÞi 

drdkOðr; kÞf ðr; kÞ ð : drdkf ðr; kÞ

ð122Þ

They are important because the first momentum moments are proportional to the density and the average velocity. When the distribution is the equilibrium FD distribution, then they have been encountered ð before, as Eq. (14), for example, can be written in the language of hknx i / knx fFD ðEðkÞÞdk for n ¼ 0. For one dimension the transverse momentum components in the numerator and denominator of Eq. (122) can be performed, resulting in the supply function encountered in Eq. (97), and allowing for the suppression of the subscript on the remaining component via kx ) k. For one dimension, then, the number density is proportional to hk0 i and current density to hk1 i. The generalization to present circumstances for a distribution function, which is spatially dependent, follows analogously: ð 1 1 rðxÞ ¼ f ðx; kÞdk 2p
1 ð1 ð123Þ 1 hk
f ðx; kÞdk J ðxÞ ¼ 2p
1 m A cautionary note about Eq. (123): as written, r and J are number density and number current density, respectively, as the charge of the carrier is not present. In the Poisson and continuity equations, for example, the charge q reappears in their coefficients. Consider three well‐known examples. First, the equipartition theorem follows from the evaluation of the second moment when the distribution function is an MB distribution via ð1 pffiffiffiffiffiffiffiffiffiffi EðkÞe
bEðkÞ dk h2 p=a3
1
1 pffiffiffiffiffiffiffiffi ¼ hEðkÞi ¼ ð 1 ; ð124Þ ¼ 2b 8m p=a e
bEðkÞ dk
1

where the parabolic relationship between E and k is used, b ¼ 1=kB T, and a ¼ b
h2 =2m. Note the (1/2) coefficient on the RHS, reflecting that Eq. (124) is a 1D evaluation: when all three dimensions are considered, hEx i þ hEy i þ hEz i ¼ 3kB T=2. Parenthetically, Eq. (124) reappears in the discussion of thermal emittance. Second, consider the case when f(x,k) is a Gaussian distribution in k only, that is,

ELECTRON EMISSION PHYSICS

f ðkÞjGaussian

(   ) ro k
ko 2 ¼ pffiffiffi exp
: Dk Dk p

49

ð125Þ

It then follows from Eq. (123) that J ¼ ð
hko =mÞr, or current density is the product of the mean (or center) velocity and number density. Third, and extending the second example, evaluate ]t r using Eq. (123) and insert it into Eq. (121) to obtain ] 1 rðx; tÞ ¼
]t 2p

1 3
h k ] F ] 4@ A f ðx; k; tÞ þ f ðx; k; tÞ5dk m ]x m ]k
1

ð1

20

] ¼
J ðx; tÞ ]x

ð126Þ

where use has been made of f ðx; 1Þ ¼ 0, allowing the ]k f term to be integrated and summarily dispensed with. Eq. (124), known as the continuity equation, is the classical distribution function version whose quantum mechanical counterpart is superficially similar but in detail a great deal more subtle, a task to which we now turn. 2. Current Density in the Schro¨dinger and Heisenberg Representations To find the quantum extensions of expressions in Eqs. (123) and (126), the bra‐ket notation requires additional formalism (Rammer, 2004). In the continuum limit and in an arbitrary number of dimensions, the kets jxi and jki satisfy hxjki ¼ ð2pÞ
d=2 expðik  xÞ hxjx0 i ¼ dðx
x0 Þ hkjk0 i ¼ dðk
k0 Þ

ð127Þ

The vector notation significantly complicates the formulas without a commensurate pedagogical benefit, so the 1D (d ¼ 1) case shall be considered henceforth, leaving the higher‐dimensional analogs to be intuited (or obtained from hardier treatments) (Reichl, 1987). An operator of particular importance is the identity operator: generalizing from when x and k are discrete, it is ð1 ð1 ^ I ¼ ð2pÞ
1=2 jxihxjdx ¼ð2pÞ
1=2 jkihkjdk: ð128Þ
1


1

50

KEVIN L. JENSEN

Evolution of the wave function is governed by the evolution (alternately, ^ ^ propagation) operator UðtÞ such that jcðtÞi ¼ UðtÞjcð0Þi. It follows from the definition of jcðtÞiand the constancy of the energy E that for a ^ then UðtÞ ^ Hamiltonian H, must satisfy ^ ^ E ¼ hcðtÞjHjcðtÞi ¼ hcð0ÞjHjcð0Þi

ð129Þ

^ 1 ÞUðt ^ 2 Þjcð0Þi ¼ Uðt ^ 1 þ t2 Þjcð0Þi jcðt1 þ t2 Þi ¼ Uðt

ð130Þ

^ UðtÞjcð0Þi ^ hcðtÞjcðtÞi ¼ hcð0ÞjUðtÞ ¼ hcð0Þjcð0Þi:

ð131Þ

h i   ^ B ^B ^ UðtÞ ^ ^ A ^ ^^ Equation (129) indicates that H; ¼ 0, where A; 
BA ^ ^ (the commutator relation). Eq. (130) indicates that UðtÞ ¼ exp f ðHÞt (where f is a function), and Eq. (131) demonstrates that U is unitary ^
1 ¼ U^{ , or the inverse of U is its adjoint). Taken together, the (i.e., U   ^ ^ h . simplest function to satisfy these requirements is UðtÞ ¼ exp
iHt=
The treatment of jcðtÞi so far has used time‐independent operators and time‐dependent wave functions, collectively known as the Schro¨dinger picture or representation. Operators in this representation are designated with an S subscript (S). An alternate approach, referred to as the Heisenberg picture or representation, designated by an H subscript ( H), makes the wave function time independent and transfers the time dependence to the operators O via ^ S UðtÞ: ^ H ðtÞ ¼ UðtÞ ^ ^ O O ^ H ðtÞ is then The variation in time of O 0 1 0 1 ] ^ ] ] ^ S UðtÞ ^ S @ UðtÞ ^ AO ^ A ^ ^ O þ UðtÞ OH ðtÞ ¼ @ UðtÞ ]t ]t ]t i i h^ ^ ¼ H ; O ðtÞ S H h


ð132Þ

ð133Þ

where Eqs. (131) and (132) have been used. Thus, in the Heisenberg representation the time variation of an operator is given by its commutation with the Hamiltonian. Let A, B, and C be operators. It is trivial to ½A; BC  ¼ h show i ^ ¼ i, it follows ½A; BC þ B½A; C . Coupled with the Heisenberg relation x^n ; k that h i ^ ¼ in^ ^n Þ: ^n ; k xn
1 ¼ i]x^ ðx ð134Þ x

ELECTRON EMISSION PHYSICS

^, it follows that Expressing f ð^ xÞ as a power series in x h i ^ ¼ i ] f ð^ xÞ: f ð^ xÞ; k ^ ]x Now consider the density operator defined by [compare Eq. (96)] X ^ nH ð t Þ ¼ f ðE ÞjcðtÞihcðtÞj; E o

51

ð135Þ

ð136Þ

from which it follows that nðx; tÞ ¼ hxj^nðtÞjxi and therefore hxj½V ð^ x Þ; ^ nðtÞjxi ¼ 0. Another way to express this is by observing that the ^ jxi vanish if O is the commutator of a position‐ diagonal elements hxjO dependent operator with the density operator. It follows that for the diagonal elements (the ‘‘off‐diagonal’’ elements of the density operator are considered in the discussion of the Wigner function) i i
o  
2 h ^ 2 h h2 ] n ^ ^ S; ^ k; ^nH ðtÞ ; k ;^ H n H ðt Þ ¼ nH ð t Þ ¼ ^ 2m 2m ]x

ð137Þ

where the anticommutator is defined by fA; Bg ¼ AB þ BA. Identify the current operator as n o ^ ^j S ¼ 1 ^ hk nS ;
2m ð138Þ n o ^ U ^ ðtÞ ^ ^ ðtÞ: ^j H ðtÞ ¼ 1 U nS ;
hk 2m Coupled with Eq. (137) and Eq. (133), Eq. (138) implies that ] ] ^ nH ðtÞ þ ^j H ðtÞ ¼ 0: ^ ]t ]x

ð139Þ

^jcðtÞi ¼ Eq. (139) is the quantum analog to Eq. (126). Using hxjk i]x cðx; tÞ, it follows that n o h
^ jcðtÞi hcðtÞj ^nS ; k j ðx; tÞ ¼ hco j^j H ð^ x; tÞjco i ¼ 2m ð140Þ  
h c{ ðx; tÞ]x cðx; tÞ
cðx; tÞ]x c{ ðx; tÞ ¼ 2mi which is the conventional relation for the current density in the Schro¨dinger picture. Consider the special case when jco i ¼ jki, that is, the wave function is a momentum eigenstate, as shown in Figure 15. The incident wave is cinc ¼ ð2pÞ
1=2 expðikxÞ, for which the incident current is jinc ¼
hk=m; as n(x,t) is a number density, j(x,t) will be the flux of particles. Current density, in the conventional sense of charge per unit area per unit time, corresponds to

52

KEVIN L. JENSEN

e ikx

t(k) e ik'x

r(k) e−ikx

FIGURE 15. Schematic representation of incident, transmitted, and reflected waves incident on a general barrier for which V( 1) ¼ 0.

qj(x,t). After interacting with a potential V ðjxj  1Þ ¼ 0, the transmitted and reflected waves are given by ctrans ¼ ð2pÞ
1=2 tðkÞexpðikxÞ and cref ¼ ð2pÞ
1=2 rðkÞexpð
ikxÞ, respectively. Thus, hk=m jtrans ¼ jtðkÞj2
jref ¼ jrðkÞj2
hk=m

ð141Þ

Conservation of particles demands that jtrans þ jref ¼ jinc or jtðkÞj2 þ jrðkÞj2 ¼ 1;

ð142Þ

which is obtained by taking the ratio of both sides with the incident current. The transmission probability for a given momentum is therefore taken as the ratio between the transmitted and incident current densities, but T ðkÞ ¼ jtðkÞj2 only for V ðjxj  1Þ ¼ F ðjxj  1Þ ¼ 0; when the RHS and the LHS differ in reference energy, or the fields are different, then changes occur in the expression for transmission probability.

3. Current Density in the Wigner Distribution Function Approach The common usage of Eq. (140) lacks resemblance to the distribution function approach of Eq. (126). In a related vein, regardless of the (visual) similarity, a difference exists between f ðx; k; tÞ and fo ðEÞjck ðxÞj2 , and so the latter is not the quantum analog of the distribution function sought. Something else is required. Wigner suggested the following function as a candidate (Hillery et al., 1984):

53

ELECTRON EMISSION PHYSICS

f ðx; k; tÞ ¼ 2

ð1
1

e2iky hx þ yj^ nðtÞjx
yidy:

ð143Þ

Combining Eqs. (137) and (139), it follows ] f ðx; k; tÞ ¼
2 ]t

8 <

ð1

h
]x^ f^nðtÞ; k^gjx
yi e2iky dy hx þ yj : 2m
1 9 = i xÞ; ^ nðtÞjx
yi þhx þ yj ½V ð^ ; h


ð144Þ

Using Eq. (136), first term of the integrand is ^ hx þ yj]x^ f^ nðtÞ; kgjx
yi ¼ kð]xþy þ ]x
y Þhx þ yj^nðtÞjx
yi ¼ 2k]x hx þ yj^ nðtÞjx
yi

ð145Þ

The second term is a bit more involved: ^Þ; ^nðtÞjx
yi ¼ ðV ðx þ yÞ
V ðx
yÞÞhx þ yj^ hx þ yj½V ðx nðtÞjx
yi ð1 1 0 e
2ik y f ðx; k0 ; tÞdk0 ¼ ðV ðx þ yÞ
V ðx
yÞÞ 2p
1

ð146Þ Introducing the concise notation V ðx; k
k0 Þ ¼


i p
h

ð1
1

0

e2iðk
k Þy fV ðx þ yÞ
V ðx
yÞgdy

ð147Þ

and combining Eqs. (144)–(147) results in the time‐evolution equation for the Wigner distribution function (WDF): ] hk ]
f ðx; k; tÞ ¼
f ðx; k; tÞ þ ]t m ]x

ð1
1

V ðx; k
k0 Þf ðx; k0 ; tÞdk0 :

ð148Þ

The immediate impact of quantum effects is the dependence of V ðx; k
k0 Þ on the potential at locations other than x; that is, the integrand in Eq. (147), is nonlocal, meaning the behavior of f at x depends on the behavior of V(x0 ) for x0 away from x. The impact of quantum effects can be related to Boltzmann’s transport equation (BTE) by expanding the integrand of Eq. (147)

54

KEVIN L. JENSEN

 2nþ1 1 X 2y2nþ1 ] V ðx þ yÞ
V ðx
yÞ ¼ V ðxÞ ]x ð 2n þ 1 Þ! n¼0

ð149Þ

and using the substitution ð1

n 2iky

y e
1

  i ] n dy ¼
2pdðkÞ: 2 ]k

ð150Þ

The generalization to the BTE is then



]
hk ] 1 X1 ð
1Þn 2nþ1 f ðx; k; tÞ : ]x V ðxÞ ]2nþ1 f ðx; k; tÞ ¼
f ðx; k; tÞ þ k 2n n¼0 ]t m ]x
h 2 ð2n þ 1Þ!

ð151Þ For potentials that are at most quadratic, all but the first term in the sum in Eq. (151) vanish. Only the field term survives, and the classical form of the BTE is satisfied by f(x,k,t). It follows that under such circumstances, f(x,k,t) ¼ f(xcl(t),kcl(t),0), where xcl(t) and kcl(t) are classical trajectories—but it does not follow that f(x,k,t) is a classical distribution function as there are regions for which f(x,k,t) can be negative, and probability distributions do not behave in such a manner (Hillery et al., 1984; Kim and Noz, 1991; Rammer, 2004; Reichl, 1987). Nevertheless, the first two moments of the WDF provide the particle number density and current density, respectively, or ð 1 1 nðx; tÞ ¼ hxj^ nðtÞjxi ¼ f ðx; k; tÞdk 2p
1 ð ð152Þ 1 1
hk f ðx; k; tÞdk Jðx; tÞ ¼ 2p
1 m ð

as can be shown using Eq. (143) and eiz dz ¼ 2pdðzÞ, which mirrors the behavior of the classical distribution function. Moreover, Eq. (139) follows from Eq. (152) and Eq. (148) and the antisymmetry of V(x,k). Examples highlighting the quantum behavior and its differences from classical distributions are given next. a. Wave Packet Spreading (No Potential). Consider a wave packet constructed by summing over plane wave states with a Gaussian weighting factor in momentum centered about k ¼ 0: it is trivial to shift the center momentum to a non‐zero value for a traveling wave packet. The normalized wave function is

55

ELECTRON EMISSION PHYSICS

8 0 12 9 11=4 ð1 < = 2 k A cðx; tÞ ¼ @ exp
@ A þ ikx
iot dk 2 : ; pDk Dk
1 8 0 1
1 9 1
1=2 0 11=4 0 = < Dk2 x2 2 1 i
h t i
h Dk @1 þ A A A @ t ¼@ þ exp
; : pDk2 Dk2 2m 2m 4 0

ð153Þ

where
hoðkÞ ¼
h2 k2 =2m. The probability density then is a Gaussian given by 0 rðx; tÞ ¼ @

11=2 0

2 A pDk2

0

12 1
1=2

ht A A @ 1 þ@
Dk4 2m

8 0 12 1
1 9 0 > > < Dk2 x2 ht A A = @ 1 þ@
exp
> > Dk4 2m 2 ; :

ð154Þ

The wave packet therefore spreads as time increases, as shown in Figure 16. Compared to Eq. (154), the Wigner formulation of the same problem is elegant. At time t ¼ 0, f(x,k,0) is given by ð 1 1 2iky f ðx; k; 0Þ ¼ e cðx þ y; 0Þcðx
y; 0Þdy 2p
1 8 0 12 9 ð155Þ < 2 1 2 2 1@ k A = ¼ 2 exp
Dk x
: 2 Dk 2 Dk ; The integrations that lead to Eq. (155) are readily employed when using Eq. (153). A regrouping of terms then results in 2

[x1(t), k1(t)] 1

k

[x2(t), k2(t)] 0

−1

−2 −3

t = 0.0 fs t = 0.2 fs

−2

−1

0 x

1

2

3

FIGURE 16. Spreading of the Gaussian wave packet in the Wigner distribution function approach. Two trajectories are shown that demonstrate the classical trajectory behavior.

56

KEVIN L. JENSEN

(     ) 2 1 2 hk 2 1 k 2
f ðx; k; tÞ ¼ 2 exp
Dk x
t
; Dk 2 m 2 Dk

ð156Þ

from which (with much effort) an integration over k reproduces Eq. (153). Examination shows that Eq. (156) can be rewritten as   hk
ð157Þ f ðx; k; tÞjV ðxÞ¼0 ¼ f x
t; k; 0 ; m where the subscript reinforces that force‐free evolution is occurring. The spreading of the wave packet is therefore tantamount to a shearing of an ellipse in phase space such that while the area of the ellipse bounded by a given contour line remains constant, it is progressively elongated (conservation of area in phase space is a notion reappearing in the discussion of emittance). Analogous to the spreading of a wave packet for an electron shown in Figure 16 (Dk ¼ 0.1 nm
1 and t ¼ 0.0 and 0.2 fs for a region 1 nm across and 1 nm
1 wide), Eqs. (153) and (156) are shown schematically in Figure 17. More can be evoked from this example. First, observe that for potentials that are at worst quadratic, Eq. (151) can be written as  ]
hk ] 1 f ðx; k; tÞ ¼
þ ]x V ðxÞ]k f ðx; k; tÞ; ð158Þ ]t m ]x
h which is seen as the BTE for one dimension. The term in curly brackets ^ For systems in which on the RHS can be treated as an operator O. the energy is constant, O^ E ¼ 0 (a restatement of Hamilton’s equations), t = 0.0 fs t = 0.2 fs t = 0.4 fs

Density [a.u.]

0.8

0.6

0.4

0.2

0 −3

−2

−1 0 1 Distance [0.1 nm]

2

3

FIGURE 17. Spreading of the Gaussian distribution density (0th moment of the distribution function) as a function of time.

ELECTRON EMISSION PHYSICS

57

then f(x,k,t) is written as f(E) and Eq. (158) is automatically satisfied along trajectories, designated by the pair of phase‐space coordinates (xp(t),kp(t)), which satisfy
2 k2p h

þ V xp ¼ E: 2m

ð159Þ

For a free particle, V(x) ¼ 0 and E ¼ ð
hk2p Þ=2m, and Eq. (158) demon
hk strates xp ðtÞ ¼ x
m t, in agreement with Eq. (157), which is what is meant when it is stated that Wigner trajectories are equivalent to classical trajectories for up to quadratic potentials. The equivalence allows for the conclusion that if V(x) is a linear function of x, or V(x) ¼ gx (the notation reflecting the common gravitational example of a linear potential), then it follows that   hk
g 2 g t ; k þ t; 0 ; ð160Þ f ðx; k; tÞ ¼ f x
t
m 2m
h which explicitly satisfies Eq. (158) and satisfies f(x,k,t) ¼ f(E(x,k)). The treatment here is more improvised than precise; Rammer (2004) and Kim and Noz (1991) provide a careful demonstration of Eq. (160). Before Wigner trajectories are dismissed as idle curiosities of a slothful imagination, they do in fact have some value: numerical simulations of particle transport are beholden to trajectories of pointlike creatures, and so introducing a classical notion into a system with quantum behavior, a ‘‘quantum trajectory’’ concept, has merit (Hsu and Wu, 1992; Jensen and Buot, 1989, 1990, 1991; Martin et al., 1999). Another quantum trajectory concept, the Bohm trajectories (Bohm and Staver, 1951; Dewdney and Hiley, 1982; Vigier et al., 1987) similarly makes a clever attempt to introduce classical trajectory concepts via Schro¨dinger’s equation itself. b. The Harmonic Oscillator. The simplicity of Eq. (156) erroneously implies that the Wigner function is a probability distribution function and therefore is positive for all values of x and k, not only because the contours act as trajectories but because f(x,k,t) acts like a classical phase‐space probability distribution function by giving momentum and current density as moments of the distribution. The Wigner function, regardless of its other virtues, is not a probability distribution; the simplest system to see how it is not (but also how quantum mechanics is intriguingly different) is the harmonic oscillator. Classically, the energy of an oscillator can be written as E¼


2 k2
h2 k2o x 2 h þ ; 2m 2m L

ð161Þ

58

KEVIN L. JENSEN

where a characteristic length (L) and momentum ð
hko Þ have been introduced. Trajectories correspond to contours of E, and therefore, xp ðtÞ ¼ xo cosðot þ fÞ kp ðtÞ ¼
xo o sinðot þ fÞ

ð162Þ

where f is a phase and
ho ¼
h2 ko =2mL 
h2 a=2m. As is generally true, it is pragmatic to know one’s final destination before embarking; to that end, a concise account of the quantum treatment of the oscillator is given to show that f(x,k,t) ¼ f(E(x,k)) in a manner foretold by Eq. (155). The unconventional representation of the energy in Eq. (161) and the introduction of a is, as expected, to simplify Schro¨dinger’s equation in operator parlance, which becomes

2  ^ þ a2 x ^2 jcn i ¼ k2n jcn i; k

ð163Þ

where the n subscript (n) distinguishes the energy levels and anticipates the conclusion that the energy of the oscillator will be quantized. The observation that

  h i ^ þ a^ ^ þ a^ ^2 þ x ^ ^2 þ ia x ^; k
ik x ik x ¼k ð164Þ ^2 þ x ^2
a ¼k where the commutator of position and momentum has been used, suggests the introduction of ‘‘creation’’ and ‘‘annihilation’’ operators (the nomenclature to become clear) defined by

 ^ þ a^ ^ x a{ ¼ a
1=2
ik

 ^ þ a^ ^ x a ¼ a
1=2 þik

ð165Þ

and satisfying h i ^ ¼2 ^; k a; ^ a{ ¼
2i x ½^ En jcn i ¼


2 a { h ð^ a a^þ 1Þjcn i 2m

ð166Þ

The similarities to the creation and annihilation operators introduced earlier in Eq. (39) are intentional, but there are important differences. Using the ½A; BC  commutation relations

ELECTRON EMISSION PHYSICS

59

2

h a { ^ a{jcn i ¼
ð^ a{jcn i H^ a a^þ 1Þ^ 2m


2 a {  { h ^ a{a^þ 1 jcn i a; ^ a þ^ a ^ 2m 8 9 <
h2 a = ¼^ a{ ð^ a{a^þ 3Þ jcn i ¼ ðEn þ
hoÞ^ajcn i : 2m ;

¼

ð167Þ

Therefore, the effect of ^ a{ onjcn i has been to raise the energy eigenvalue by 2 h a=m ¼

ho; that is, En ¼ n
ho þ E0 . Normalization is resolved by starting with the ground state   ] hxj^ ajc0 i ¼ 0 ¼
þ ax c0 ðxÞ; ð168Þ ]x for which the solution is c0 ðxÞ ¼

a1=4 p

  1 2 exp
ax : 2

ð169Þ

The action of the Hamiltonian on Eq. (169) identifies E0 ¼
h2 a=2m ¼
ho=2. Let jcn i ¼ Nn ð^ a{Þn jc0 i, where Nn is the normalization found by insisting that 1 ¼ hcn jcn i ¼ ðNn =Nn
1 Þ2 hcn
1 j^ a^ a{jcn
1 i ¼ ðNn =Nn
1 Þ2 hcn
1 j^ a{a^þ 2jcn
1 i

ð170Þ

¼ ðNn =Nn
1 Þ2 hcn
1 j2ðn
1Þ þ 2jcn
1 i ¼ ðNn =Nn
1 Þ2 2n It follows thatNn ¼ ð2n n!Þ
1=2 ða=pÞ1=4 . The solution to the harmonic oscillator is now complete, and the representation in the jxi basis can be obtained from the definition of the ground state hxj^ ajc0 i ¼ a
1=2 ð]x þ axÞc0 ðxÞ ¼ 0; the normalized solution is c0 ðxÞ ¼

a1=4 p

  1 exp
ax2 : 2

ð171Þ

ð172Þ

The use of Eq. (172) to find the ground‐state Wigner function, while certainly a candidate for consideration, is not a particularly compelling one because it does not differ in appearance much from the wave packet example. Higher n wave functions are of greater pedagogical interest, but they require further work. Let

60

KEVIN L. JENSEN

  1 cn ðxÞ ¼ Nn hðxÞexp
ax2 ; 2 where h(x) is to be determined. From ^ a{jcn
1 i,      1 2 1 2
1=2 ð
]x þ axÞ hn
1 ðxÞexp
ax hn ðxÞexp
ax ¼ a : 2 2

ð173Þ

ð174Þ

It follows that hn ðxÞ ¼ a
1=2 ð
]x hn
1 ðxÞ þ 2axhn
1 ðxÞÞ;

ð175Þ

where h0(x) ¼ 1. Eq. (175) is a variation of the recurrence relation for Hermite polynomials (Abramowitz and Stegun, 1965), namely, Hnþ1 ðyÞ ¼
]y Hn ðyÞ þ 2yHn ðyÞ and so for the nth level,  

a1=4

1 cn ðxÞ ¼ ð2n n!Þ
1=2 Hn a1=2 x exp
ax2 : ð176Þ p 2 Eq. (176) is what was sought to find the nth‐level Wigner function. Straightforward evaluation of Eq. (155) but with Eq. (173) shows that for n ¼ 0 and 1 2 3 2

1 f0 ðx; k; 0Þ ¼ exp4
k þ a2 x2 5 a 3 2 3 2 ð177Þ 2



2 1 f1 ðx; k; 0Þ ¼ 4 k þ a2 x2
15exp4
k2 þ a2 x2 5 a a The higher n Wigner functions are evaluated analogously. The classical orbits for the harmonic oscillator are xðtÞ ¼ ðe=aÞ1=2 cosðot þ ’Þ and kðtÞ ¼ ðaeÞ1=2 sinðot þ ’Þ, where ’ is specified by initial conditions and e ¼ 2E=ð
hoÞ. It is easily shown that f0 ðxðtÞ; kðtÞÞ ¼ e
e f1 ðxðtÞ; kðtÞÞ ¼ ð2e
1Þe
e

ð178Þ

In other words, as suggested, the classical trajectories correspond to the contour lines of the Wigner function. The exploitation of the trajectory concept is of more than just pedagogical interest; it may, in fact, serve as a bridge between quantum (Wigner function) and classical (BTE) simulations (Hsu and Wu, 1992; Jensen and Buot, 1989, 1991; Jensen and Ganguly, 1993; Martin et al., 1999; Vigier et al., 1987). Some care is required, however, as Eq. (178) contains an additional feature: unlike a classical distribution function, the Wigner function can assume negative values, as is apparent when e < 1 for f1(x,k) (Figure 18), a feature that prevents its interpretation

ELECTRON EMISSION PHYSICS

61

f1(x,k) > 0

f1(x,k) < 0 FIGURE 18. The Wigner distribution function f1(x,k) for the n = 1 harmonic oscillator.

as a true probability distribution function, regardless of the utility of its moments for the evaluation of the number and current densities. Less timorous spirits have tackled the concept of negative probability head on (Feynman, 1987), but here, such a feature is a quixotic artifact of an otherwise useful approach. c. The Gaussian Potential Barrier. A final pedagogically hvaluable i example is the case where V(x) is of the form V ðxÞ ¼ Vo exp
ðx=DÞ2 . A blessedly short derivation then shows that ð i 1 2iky e ½V ðx þ yÞ
V ðx
yÞdy V ðx; kÞ ¼
p
h
1 ð179Þ   2D ¼
pffiffiffi Vo exp
D2 k2 sinð2kxÞ p
h Eq. (179) is curious and useful not because real potentials are Gaussian (they are not) nor because the Wigner function is easily evaluated (it is not) but because consideration of the Gaussian potential provides a relatively clear distinction between the regimes where through‐the‐barrier (tunneling) versus over‐the‐barrier (thermionic emission) dominate without the necessity of indulging gradient expansions characteristic of Eq. (151). The argument is simple: if D is large, indicating that the barrier is wide, then the exponential in 0 Eq. (179) is sharply peaked, so that only values for which k  k survive in Eq. (148) so that because of the sine term, all the even terms of a Taylor expansion including the 0th‐order term vanish, leaving the classical equation of Eq. (121). Conversely, if D is small, then the exponential term is broad and momentum values far from k0 contribute. Quite generally, then, quantum effects are unimportant when kx oscillates many times over a length scale characteristic of the barrier width at E(k), but are important when kx

62

KEVIN L. JENSEN

wiggles only a few times over the characteristic length scale. The value of k is for the largest value appreciably present, which, for field emission from metals is the Fermi value kF, whereas x is of order D. Transplanting this newly acquired intuition to the field emission barrier, but ignoring the image charge term, the potential barrier for an applied field F has a thickness of F/F at an incident energy equal to the chemical potential. Thus, tunneling is important when kF

F F pffiffiffiffiffiffiffiffiffi  2p ) F  2mm: F 2p
h

ð180Þ

For example, using values characteristic of copper (m ¼ 7 eV, F ¼ 4.6 eV), then F is on the order of 10 eV/nm, or equivalent to 10 GV/m. In practice, fields between 4 GV/m and 8 GV/m produce appreciable tunneling current from metals when the work function is several electron volts.

4. Current Density in the Bohm Approach An approach to the evaluation of current density and the transmission probability due to Bohm and Hiley (1985) is a natural introduction to the Wentzel–Kramers–Brillouin (WKB) methods used below [Dicke and Wittke (1960); although, as emphasized by Forbes (1968), given modern usage epitomized by Murphy and Good (1956) the designation Jeffreys‐Wentzel‐ Kramers‐Brillouin (JWKB) is perhaps preferred]. The wave function is represented at cðxÞ ¼ RðxÞexp½iSðxÞ, where R and S are real functions. When inserted into the time‐dependent Schro¨dinger equation i
h] t c ¼

!
2 2 h ] þ V ðxÞ c;
2m x

ð181Þ

then Eq. (181) becomes, on separating the real and imaginary parts to the LHS, respectively, 8 9 < i =
2 h h
h2  2
1 2 2
1 2 2ð]x RÞð]x S Þ þ R]x S ð]x S Þ
R ]x R þ kv þ
h]t S ¼ i
hR ]t R þ : ; 2m 2mR

ð182Þ

whereV ðxÞ ¼ ð
hkv ðxÞÞ2 =2m. Because R and S are real, each side of Eq. (182) separately is zero. The LHS gives the Hamilton–Jacobi equation (Goldstein, 1980) if
h]x S is identified as a momentum (i.e., KE ¼ ð
h]x SÞ2 =2m is the kinetic energy) and a quantum potential (distinct from the classical potential V(x)) is defined by

ELECTRON EMISSION PHYSICS

OðxÞ ¼



2
1 2 h R ]x R; 2m

63

ð183Þ

in which case the LHS of Eq. (182) becomes
h]t S þ

i
h2 h ð]x S Þ2
R
1 ]2x R þ k2v ¼
h]t S þ KE þ OðxÞ þ V ðxÞ ¼ 0: 2m

ð184Þ

The quantum potential and the potential V(x) can be used to chart the dynamics of particles following Bohm trajectories (Dewdney and Hiley, 1982; Vigier et al., 1987; analogous to the Wigner trajectories), and has been used to provide a trajectory interpretation to the interaction of wave packets with general barriers and resonant tunneling diode (RTD) barriers. The current density is obtained by inserting the wave function into Eq. (140) to yield   h 2
]x SðxÞ ; JðxÞ ¼ RðxÞ ð185Þ m which, when compared to jðxÞ ¼ ð
hk=mÞrðxÞ, matches the interpretation of RðxÞ2 as the number density and ð
h=mÞ]x S as the velocity. The RHS of Eq. (182) reproduces the continuity equation 8 9 < 2  = i

h h ¼ i
hR
1 ]t R þ ð] r þ ]x J Þ ¼ 0: ð186Þ 2ð]x RÞð]x SÞ þ R]2x S : ; 2r t 2mR Equations (184) and (186) are beautiful, but unlike the Wigner trajectory case, evaluating the Bohm trajectories for situations even as appealing as the harmonic oscillator serves no further pedagogical value. Rather, in the development of the emission equations, the Bohm approach is useful as a backdoor approach to defining the most widely used approach to the derivation of current density for potentials encountered in electron emission, namely, the area under the potential approach to evaluating the transmission coefficient. In this situation, the time‐independent Schro¨dinger equation for momentum eigenstates is used


]2x þ kv ðxÞ2 ck ðxÞ ¼ k2 ck ðxÞ; ð187Þ where EðkÞ ¼ ð
hkÞ2 =2m, which, upon incorporating Eq. (185) to render in terms of J(x), becomes   2r m ðE
V
OÞ
J 2 : ]x J ¼ i ð188Þ
h
hr

64

KEVIN L. JENSEN

Because of the continuity equation and presumed time independence, the LHS is zero. From the definition of J, it follows that rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m ]x S ¼ i ðV ðxÞ þ OðxÞ
E Þ: h2


ð189Þ

The motive for extracting the ‘‘i’’ explicitly is because of the particular interest in the case where the electron energy is below the potential maximum so that tunneling occurs. The integral of Eq. (189) is between the zeros of the integrand. Of particular interest is the case when the density is slowly varying, which is generally (but not always) associated with a slow variation in the potential. If the density is exponentially decaying over some length scale l, then the quantum potential is of the order O 

h2 =2ml2 , and therefore small if l is large. With the designationð
hkðxÞÞ2 =2m ¼ V ðxÞ
E, a good approximation to c is then  ðx  cðxÞ / kðxÞ
1=2 exp
kðx0 Þdx0 ;

ð190Þ

where the coefficient follows from Eq. (185). As introduced previously, the transmission coefficient T(E) is the ratio between the current transmitted through the barrier and the incident current. If the transmitted wave function is a plane wave whose magnitude is decreased as per Eq. (190), it then follows that  TðEÞ  exp
2

ð xþ

0

0



kðx Þdx ;

ð191Þ

x


where the limits of the integral are the zeros of the integrand. Eq. (191) is the form most commonly invoked in the determination of tunneling current (e.g., field emission), even though its most serious defect is a neglect of the momentum dependence of the coefficient of the exponential term, approximating it rather by unity. Still, the exponential term in Eq. (191) captures the dominant features and, coupled with other approximations, enables tractable analytical solutions for classes of potentials of particular importance here. Another problem is the behavior of Eq. (191) near the barrier maximum, where the neglect of O(x) is an issue. There is therefore pedagogical value to examine exactly solvable cases to find how well Eq. (191) holds—and conversely, where it fails and how it should be modified.

ELECTRON EMISSION PHYSICS

65

B. Exactly Solvable Models Classes of potentials whose simplicity or particular features facilitate methods for which numerical evaluation of Schro¨dinger’s equation is not needed are considered next. The first class is those in which Schro¨dinger’s equation is exactly solvable (the square barrier and triangular barrier potentials) for which the general methodology is to find basis states that are analytically tractable. The second class is those for which the integral in the area under the curve (AUC) method suggested by the Bohm analysis is analytic.

1. Wave Function Methodology for Constant Potential Segments The general technique of the wave function methodology was encountered in the discussion of Eq. (35), but in the present analysis, considerably more attention to its detail is useful. Reconsider the 1D Schro¨dinger equation in position space, that is, hxjcE i ¼ cE ðxÞ for which (

)  
2 ] 2 h þ V ðxÞ cE ðxÞ ¼ EcE ðxÞ
2m ]x

ð192Þ

for potentials that are at worst piece‐wise discontinuous for a finite number of regions, that is, for n ¼ 1. . .N, lim jV ðxn þ dÞ
V ðxn
dÞj < 1:

d!0

ð193Þ

As the energy is also finite, Eqs. (192) and (193) therefore require that the second derivative also be at most piece‐wise discontinuous, and therefore, that the wave function, as well as its first derivative, is continuous, conditions formally expressed as lim jcE ðxn þ dÞ
cE ðxn
dÞj ¼ 0

d!0

0

0

lim jc E ðxn þ dÞ
c E ðxn
dÞj ¼ 0

ð194Þ

d!0

where prime (0 ) indicates derivative with respect to argument. Momentum eigenstates are convenient for the evaluation of current, and they are henceforth exclusively used. The wave functions in Eq. (194) are superpositions of positive and negative momentum states, both of which have the same energy eigenvalue (a consequence of the parabolic relationship between E and k). For the special case where the potential for xn  x  xnþ1 is constant and equal to Vn,

66

KEVIN L. JENSEN

cn ðxÞ ¼ tn expðik xÞ þ rn expðffi
ikn xÞ pnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hkn ¼ 2mðE
Vn Þ


ð195Þ

where t and r are complex coefficients for waves moving to the right and left, respectively. A matrix representation of Eq. (194) therefore suggests itself (anticipated by the Kronig–Penney treatment)      cn ðxÞ expð
ikn xÞ expðikn xÞ tn ðkÞ ¼ ð196Þ ]x cn ðxÞ ikn expðikn xÞ
ikn expð
ikn xÞ rn ðkÞ The coefficients’ vector shall be designated byzn ðxÞ, and the 2 2 matrices byMn ðxÞ. Matching wave function and first derivative entails





Mn
1 ðxn Þ zn
1 ðxn Þ ¼ Mn ðxn Þ zn ðxn Þ:

ð197Þ

Solving introduces matrices given by (Brennan and Summers, 1987; Jensen 2003b; Tsu and Esaki, 1973; Vassell et al., 1983): ^ SðnÞ  Mn
1 ðxn Þ
1 Mn ðxn Þ ( ðkn þ kn
1 Þexp½iðkn
kn
1 Þxn  1 ¼ 2kn
1 ð
kn þ kn
1 Þexp½iðkn þ kn
1 Þxn 

ð
kn þ kn
1 Þexp½
iðkn þ kn
1 Þxn 

)

ðkn þ kn
1 Þexp½
iðkn
kn
1 Þxn 

ð198Þ For a potential separated into N regions, subject to t0 ¼ 1, r0 ¼ r, tN ¼ t, and rN ¼ 0, that is, the incident wave on the left is normalized to unity and there is no wave incident from the RHS, then )     (Y N t 1 ^ : ð199Þ ¼ SðnÞ 0 r



n¼1

By virtue of the fact that no wave is incident from the right, Eq. (199) therefore indicates tðkÞ ¼

8" < Y N :

n¼1

^ SðnÞ

# 9
1 = 1;1

;

;

ð200Þ

where the (1,1) subscript (1,1) indicates that the first‐row, first‐column entry of the matrix is given by the product of S(1) through S(N). From the definition of current density given by Eq. (140), it follows that the incident, reflected, and transmitted currents are, respectively,

67

ELECTRON EMISSION PHYSICS

jinc ðkÞ ¼


k h m

jref ðkÞ ¼


hk jrðkÞj2 m

jtrans ðkÞ ¼


k h jtðkÞj2 m

ð201Þ

when k0 ¼ kN ¼ k the transmission coefficient T(k), representing the ratio of the transmitted current with the incident current, is as beforejtðkÞj2 . When kN 6¼ k, thenTðkÞ ¼ ðkN =kÞjtðkÞj2 . The simplest case of a (N ¼ 1) step function V(x) ¼ Vo for x 0 and 0 otherwise, for example, results in tðkÞ ¼

2k ) k þ k1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E ðE
Vo Þ TðkÞ ¼ ¼ pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðk þ k1 Þ E þ ðE
V o Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the form of T(k) is valid only for
hk >
hk1 ¼ 2mðE
Vo Þ. 4k1 k

ð202Þ

2. The Square Barrier The next level of complexity is a simple square barrier of heightVo ¼
h2 k2v =2m, for N ¼ 2, such 1=2 that k0 ¼ k2 ¼ k, x0 ¼ 0, and x1 ¼ L. Consequently, k1 ¼ k2
k2v is real or imaginary, depending on whether the E > Vo or E >
ðk < kv Þ > 2 > < ðik
kÞ ekL
ðik þ kÞ2 e
kL ð203Þ tðkÞ ¼ 4kkeikL > > >
ð k > k Þ v > : ðk
kÞ2 eikL
ðk þ kÞ2 e
ikL

T ðk Þ ¼

8 4k2 k2 > > > > < 4k2 k2 þ ½ðk2 þ k2 ÞsinhðLkÞ2

ðk < k v Þ

4k2 k2 > > > > : 4k2 k2 þ ½ðk2
k2 ÞsinðLkÞ2

ðk > k v Þ

ð204Þ

˚ and A representative case of Eq. (204) is shown in Figure 19 for L ¼ 5 A Vo ¼ 10 eV. Also shown are asymptotic (‘‘approximate’’) limits given by

68

KEVIN L. JENSEN

80 1 2 > > > 2k > @ A expð
2kLÞ > > > < k Tapprox ðkÞ  0 12 > > > > > @ 2kk A > > : k2 þ k2

ðk < kv Þ ð205Þ ðk > kv Þ

where, for k > kv, the lower limit line replaces sin(Lk) by 1 (the upper limit is self‐evidently unity). Several observations are forthcoming. First, the approximate solution, reminiscent of the AUC WKB approach, is reasonably good for values of momentum below the barrier value kv—reasonably good, (a) Transmission coefficient

1.0 0.8 0.6 0.4 T(E)

0.2

Approx (EVo)

0

Transmission coefficient

(b)

0

5

10 15 Energy [eV]

20

25

100 10−1 10−2 10−3 10−4

T(E) Approx (E < Vo)

10−5 10−6

Approx (E > Vo)

0

5

10 15 Energy [eV]

20

25

FIGURE 19. Transmission probability (thick black line) for a rectangular barrier of height 10 eV and of width 0.5 nm. The thin dashed and solid lines are for the two limiting cases shown in Eq. (205). (b) Same as (a), but on a log scale.

ELECTRON EMISSION PHYSICS

69

that is, when E is well below Vo, in contrast to near the barrier maximum, where the approximation degrades as expected from the behavior of Eq. (204). Second, lnðTðkÞÞis approximately linear with respect to E(k) for narrow ranges in the vicinity of E ¼ m. Third, at the barrier maximum (as well as particular momentum values above it), the transmission coefficient is not unity. These observations will have bearing on the emission equations developed for general potentials in what follows.

3. Multiple Square Barriers The ‘‘area‐under‐the‐potential’’ method of evaluating the transmission coefficient can be approximated by using the trapezoidal approximation to evaluating integrals as in ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð xmax v u u2m t 2 ðV ðxÞ
E Þdx
lnfTðEÞg ¼ h
xmin N pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiX 2m Dxn V ðxn Þ
E  h
n¼1

ð206Þ

Not surprisingly, Eq. (206) looks very much like a sequence of square barriers whose cumulative effect is the product of their respective ffi transqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

mission coefficients Tn, where
lnfTn ðEÞg ¼ Dx 2m=
h2 ðV ðxn Þ
E Þ. Superficially, it appears that T(E) would not be different if the barriers were adjacent or separated by a distance. The wave nature of the electron, however, renders that conclusion inaccurate. When the barriers are far enough apart to allow a resonant level for values of the energy below the barrier maximum, then T(E) can approach unity for particular energy levels. When there are many barriers the Kronig–Penney model is approached. The opposite limit of but two barriers gives the case considered by Esaki and Tsu (Tsu and Esaki, 1973) in their analysis of the RTD. The methodology of Eq. (200) reveals the subtlety nicely: in Figure 20 the effect of repeatedly doubling the number of barriers on T(E) clearly shows the development of ‘‘bands’’ for energies above the barrier maximum of Vo ¼ 10 eV. Similarly, in Figure 21 for T(E), for which the energy range is generally below the barrier maximum, the intuition motivated by Eq. (206) accounts for much, but not all, of the behavior of T(E), in that if Tn ðEÞ is the transmission probability for n barriers, then T2n ðEÞ  ½Tn ðEÞ2  ½T1 ðEÞ2n for 2n barriers; that is, doubling the number of barriers tends to square the transmission probability (except near resonances).

0.8

Transmission coeff.

Transmission coeff.

(b)

1.0

0.6 0.4 0.2

Step function Single barrier

0

0.8 0.6 0.4 0.2

1 Barrier 2 Barriers

10

15 Energy [eV]

20

25

5

(d)

1.0

Transmission coeff.

0.8 0.6 0.4 0.2 2 Barriers 4 Barriers

0

10

15 Energy [eV]

20

25

1.0 0.8 0.6 0.4 0.2 0

4 Barriers 8 Barriers

5 10 15 20 25 15 20 25 Energy [eV] Energy [eV] FIGURE 20. (a) Step function versus single barrier transmission probability (barrier height = 10 eV). (b) Same as (a) but for single and double barriers. (c) Same as (b) but for double (two) and four barriers. (d) Same as (c), but for four and eight barriers. Evidence of bandlike formation is becoming discernible.

5

10

KEVIN L. JENSEN

Transmission coeff.

1.0

0 0

(c)

70

(a)

71

ELECTRON EMISSION PHYSICS

(a)

100

Transmission coeff.

10−2 10−4 10−6 10−8 10−10 10−12 10−14

1 2 4 8

10−16 10−18 10−20

5

6

7

8 9 Energy [eV]

10

Barrier Barriers Barriers Barriers

11

12

(b) 100

{T(E)}1/n

10−1 10−2 10−3 1 2 4 8

10−4 10−5

5

6

7

8 9 Energy [eV]

10

Barrier Barriers Barriers Barriers

11

12

FIGURE 21. (a) Summary of Figure 20 on a log scale, but showing the existence of a previously indiscernible resonance level at 8 eV. (b) Same as (a), but with the transmission probability take to the root of the number of barriers n—as expected, the area under the curve exponential factor is largely seen to govern the behavior away from resonance.

4. The Airy Function Approach Returning to the step function barrier, consider the case where, instead of being constant, the barrier is of the form V(x) ¼ Vo – Fx, where F is the product of the electric field and the electron charge. Retaining the notation Vo ¼ ð
hko Þ2 =2m and introducing F ¼
s
h2 f =2m (note that f has 2 ˚ ] and is assumed positive), Schro¨dinger’s equation becomes units of [1/A


]2x ck ðxÞ þ k2o
k2 þ sfx ck ðxÞ ¼ 0; ð207Þ

72

KEVIN L. JENSEN

where s indicates the inclination of the field, or
1 for descending, þ1 for ascending: for a triangular barrier, s is therefore (
1). Such an awkward notation may appear at best to be feigned madness, but there is method to it: extra work now will be well worth the investment later. Introduce 2 2 zðxÞ  f
2=3 jko
k þ sfxj

c ¼ sign k2o
k2 þ sfx

ð208Þ

for which Eq. (207) becomes Airy’s differential equation ]2z c
c2 zc ¼ 0 ck ðxÞ ¼ aAiðc2 zðxÞÞ þ bBiðc2 zðxÞÞ

ð209Þ

where c ¼ 1,i, ]x z ¼ c2 sf 1=3 , and a and b are arbitrary constants determined by boundary conditions (c should not be confused with the speed of light, and a and b are not to be identified with an and bn below). Note that s is apparently hidden, as when it does appear, it does so as s2 ¼ 1, but s will return below. Although Eq. (209) is correct, its utility is compromised by hiding behind the Airy functions and thereby obscuring the smooth transition to the field‐free case where the wave function become plane waves. An approach that explicitly calls out the asymptotic behavior of the Airy functions is numerically advantageous. Recalling the Bohm analysis leading to Eq. (190), a reasonable ansatz to the wave function is 8 9 <2 = Fc ðzÞ
1=4 Ziðc; zÞ ¼ pffiffiffi z exp cz3=2 :3 ; 2 p

1



1 
ðc
1Þ c2 þ 2c þ 3 Ai c2 z þ ðc þ 1Þ 3c2
2c þ 1 Bi c2 z 4 4

ð210Þ

where the new function Fc(z) (not to be confused with the field F) reflects the desire to move beyond the estimates leading to Eq. (190), and the second line (sans the peculiar coefficients) corresponds to a more traditional method of representing the wave function for a linear potential. Eq. (210) introduces and defines the Zi functions—so named to emphasize their connection to Ai(z) and Bi(z). In fact, ck(x) is linear combinations of the Zi functions for c2 ¼ 1, but it is easer to treat one case at a time, an approach that presents no difficulty provided at the end one is mindful of the shortcut. Inserting Eq. (210) into the Airy differential equation gives for F(z) ( )  2 3=2

] 2 ] þ 5 Fc ðzÞ ¼ 0: 16z þ 8z 4cz
1 ð211Þ ]z ]z

ELECTRON EMISSION PHYSICS

73

Introducing a change of variables given by zðzÞ ¼ z
3=2 , Eq. (211) becomes (

)  2 ] ] 36z þ 24ð3z
2cÞ þ 5 Fc ðzðzÞÞ ¼ 0 ]z ]z 2

ð212Þ

Inspection of Eq. (212) suggests that Fc(z) is a polynomial in z and that its adjacent coefficients (which depend on c) are related. Inserting X1 Fc ðzðzÞÞ ¼ a zn into Eq. (212) and setting the coefficients of different n¼0 n powers of z to 0 gives for the an an ð6n
1Þð6n
5Þ ; ¼ s3 48n an
1

ð213Þ

or 0 1 ð
sÞn 1 1 an ¼ pffiffiffi 2n G@3n þ Aa0 2 p 3 ð2nÞ! n p ffiffiffi



o 1 ðs þ 3Þ s2 þ 1 þ 2 2ðs
1Þ s2
1 a0 ¼ 4

ð214Þ

where the values of a0 are determined from the asymptotic expansions of the Airy functions compared to Eq. (210). As pleasing (and well‐known; Hochstadt, 1986) as the expansion entailed by Eq. (214) is, it is numerically unusable because the coefficients eventually dominate z
3n/2 and the terms fail to converge. Eq. (210), though appearing to be useful, instead hides its computational limitations behind the allure of its simplicity. Tunneling calculations routinely encounter exponentials of large terms, and their computation therefore results in machine precision limitations even when using widespread and useful numerical packages such as IMSL or LAPACK (Linear Algebra PACKage), unless care is taken to partition the calculation to appropriate regions. To appreciate what works numerically, there is value in showing what does not work. At first glance, a naı¨ve approach is to adopt a polynomial fit, with z ¼ z
3/2 of the form Fc ðzðzÞÞ ¼

X4 n¼0

bn ðcÞzn

ð215Þ

where the bn are the nth row of a b vector determined from solving the matrix equation

74

KEVIN L. JENSEN



^ b ¼ Fc C h i k
1 ^ ¼ zj C j;k

½Fc j ¼ Fc z zj

ð216Þ

and zj ¼ (j
1)/4 for a fourth‐order polynomial, 0  j  4, and where Fc(zj) can be determined from a table of Airy functions (Abramowitz and Stegun, 1965) and the relation

0 1 pffiffiffi 1=4 2 Fc ðzÞ ¼ 2 pz exp@
cz3=2 A 3 2 3 2

2 1 2

2

1 4
ðc
1Þ c þ 2c þ 3 Ai c z þ ðc þ 1Þ 3c
2c þ 1 Bi c z 5 4 4

ð217Þ

In the case of a fourth‐order polynomial fit 1 0 1 1 0 1:4142  1:4142i 1:00000 2:00000 B
0:10232 C B 0:10047 C B
0:15439  0:15237i C C B C B C B C B C C B bð1Þ ¼ B B 1:12626 C; bð
1Þ ¼ B 0:06402 C; bðiÞ ¼ B
0:06408  0:17965i C @
0:03933 A @
1:79134 A @ 0:11339  0:10189i A
0:04509  0:02511i 0:01176 0:76213 0

ð218Þ

where the argument of the b vector is the value of c. For c ¼ i, the relation Fi (z){ ¼ F‐i (z) is used. Tunneling calculations using the approximation of Eq. (217) for arbitrary barriers are generally good to 1%, depending on the potential examined; a consequence is that the numerical estimation of the transmission probability can exceed unity by a small amount for energies above the barrier maximum. Eq. (215) sacrifices more than aesthetic beauty for computational simplicity, it sacrifices accuracy: for large arguments, small discrepancies give rise to cumulative errors. This leads (begrudgingly) to the final, and workable, approach based on interpolating between known values (Tables 5 and 6). The form of Eq. (218) and the dependence of Eq. (212) on c motivate defining Fc by the real functions X and X0 as per Fc ðzðzÞÞ ¼ Xc ðzÞ  cX 0c ðzÞ:

ð219Þ

The behavior of X and X0 are shown in Figure 22 as well as tabulated in Table 6. Closely related to the Fc functions are the Hc functions, needed for derivatives of the Zi functions, defined according to pffiffiffi ]z Ziðc; zÞ ¼ zHc ðzðzÞÞZiðc; zÞ ð220Þ From Eqs. (212)–(214), it follows that Hc(0) ¼ c. Analogous to Eq. (219), introduce the real functions Wc and Wc0 defined according to

75

ELECTRON EMISSION PHYSICS TABLE 5 0 AIRY POLYNOMIAL FUNCTIONS I: VALUES OF XC z AND X C (z) FOR c ¼ 1 AND i z

X1

Xi

X0 1

X0 i

0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600 0.625 0.650 0.675 0.700 0.725 0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975 1.000

1.50000 1.50138 1.50293 1.50465 1.50658 1.50875 1.51121 1.51400 1.51712 1.52053 1.52418 1.52797 1.53183 1.53568 1.53945 1.54308 1.54654 1.54979 1.55280 1.55557 1.55809 1.56036 1.56237 1.56414 1.56566 1.56696 1.56804 1.56890 1.56957 1.57005 1.57035 1.57049 1.57047 1.57031 1.57001 1.56958 1.56903 1.56838 1.56762 1.56677 1.56583

1.41421 1.41047 1.40658 1.40259 1.39851 1.39436 1.39017 1.38594 1.38170 1.37745 1.37321 1.36898 1.36477 1.36058 1.35643 1.35231 1.34823 1.34420 1.34021 1.33626 1.33236 1.32850 1.32469 1.32093 1.31722 1.31356 1.30994 1.30637 1.30285 1.29938 1.29595 1.29256 1.28923 1.28593 1.28268 1.27947 1.27631 1.27318 1.27010 1.26705 1.26405

0.50000 0.50394 0.50794 0.51204 0.51627 0.52066 0.52528 0.53017 0.53533 0.54074 0.54632 0.55201 0.55772 0.56337 0.56891 0.57427 0.57942 0.58433 0.58897 0.59334 0.59743 0.60123 0.60475 0.60800 0.61099 0.61372 0.61621 0.61846 0.62050 0.62232 0.62395 0.62539 0.62666 0.62776 0.62871 0.62952 0.63019 0.63074 0.63117 0.63148 0.63170


1.41421
1.41781
1.42127
1.42453
1.42763
1.43054
1.43328
1.43585
1.43825
1.44049
1.44257
1.44451
1.44631
1.44799
1.44954
1.45098
1.45230
1.45353
1.45466
1.45570
1.45665
1.45753
1.45832
1.45905
1.45971
1.46031
1.46085
1.46133
1.46176
1.46214
1.46248
1.46276
1.46301
1.46322
1.46339
1.46352
1.46362
1.46368
1.46372
1.46373
1.46371

76

KEVIN L. JENSEN TABLE 6 AIRY POLYNOMIAL FUNCTIONS II: VALUES OF WC z AND W 0C (z) FOR c ¼ 1 AND i

z

W1

Wi

W0 1

W0 i

0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 0.525 0.550 0.575 0.600 0.625 0.650 0.675 0.700 0.725 0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975 1.000

0.00000
0.00625
0.01253
0.01885
0.02526
0.03178
0.03848
0.04540
0.05255
0.05987
0.06732
0.07482
0.08229
0.08965
0.09686
0.10386
0.11062
0.11712
0.12334
0.12928
0.13493
0.14030
0.14540
0.15023
0.15481
0.15914
0.16323
0.16711
0.17077
0.17424
0.17751
0.18061
0.18354
0.18632
0.18895
0.19143
0.19379
0.19602
0.19813
0.20013
0.20203

0.00000
0.00625
0.01247
0.01865
0.02478
0.03083
0.03680
0.04268
0.04847
0.05415
0.05974
0.06522
0.07061
0.07589
0.08107
0.08616
0.09116
0.09606
0.10087
0.10559
0.11023
0.11479
0.11926
0.12366
0.12799
0.13224
0.13642
0.14054
0.14459
0.14857
0.15249
0.15636
0.16016
0.16391
0.16760
0.17124
0.17483
0.17837
0.18186
0.18530
0.18870

1.00000 0.99990 0.99961 0.99910 0.99837 0.99739 0.99610 0.99448 0.99253 0.99029 0.98783 0.98523 0.98258 0.97995 0.97740 0.97499 0.97274 0.97069 0.96885 0.96723 0.96584 0.96467 0.96373 0.96299 0.96247 0.96213 0.96199 0.96202 0.96222 0.96257 0.96307 0.96371 0.96447 0.96536 0.96635 0.96746 0.96865 0.96994 0.97131 0.97276 0.97429

1.00000 1.00010 1.00039 1.00086 1.00151 1.00233 1.00329 1.00439 1.00562 1.00696 1.00839 1.00992 1.01153 1.01321 1.01495 1.01676 1.01861 1.02051 1.02245 1.02442 1.02643 1.02846 1.03052 1.03261 1.03471 1.03682 1.03895 1.04110 1.04325 1.04541 1.04758 1.04976 1.05194 1.05412 1.05631 1.05850 1.06069 1.06288 1.06507 1.06726 1.06944

(a) 1.58

1.40

X'1 X1

1.56

−1.41

(b)

0.68

c=i

−1.42

0.64

−1.43

1.52

Xi(z)

0.60

1.32

0.56

0

0.2

0.4

0.6

0.8

−1.46

1.28

0.52

c=1 1

0

0.2

0.4

z (c)

0.6

0.8

1.00

(d)

0.00

1.08

c=1

0.98

−0.16

0.97

−0.20 0.6

0.8

1

1.06

−0.08 1.04

−0.12 −0.16

1.02

−0.20

0.96

0

z

W'i (z)

W'1

−0.12

Wi(z)

W1

−0.08

Wi

Wi' W'i(z)

W1(z)

−0.04

0.99

0.4

−1.47

c=i

−0.04

0.2

1

z

0.00

0

−1.45 ELECTRON EMISSION PHYSICS

1.50

−1.44

Xi X'i

X'i(z)

1.54

X1' (z )

X1(z )

1.36

0.2

0.4

0.6

0.8

1

1.00

z

FIGURE 22. (a) Behavior of the Airy coefficients X1 and X 1 introduced in Eq. (219). (b) Behavior of the Airy coefficients Xi and X0 i introduced in Eq. (219). (c) Behavior of the Airy coefficients W1 and W0 1 introduced in Eq. (220). (d) Behavior of the Airy coefficients Wi and W0 i introduced in Eq. (221).

77

0

78

KEVIN L. JENSEN

Hc ðzÞ ¼ Wc ðzÞ  cW 0c ðzÞ:

ð221Þ

The behavior of W and W0 are shown in Figure 22 and tabulated in Table 6. Although the X and W appear to be polynomials, they are not. In practice, then, the usage of tabulated values of X, X0 , W, and W0 and interpolation to intermediate values of z via

1 1 Xc ðzn
1 < z < znþ1 Þ ¼ yð1
yÞXc ðzn
1 Þ þ 1
y2 Xc ðzn Þ þ yð1 þ yÞXc ðznþ1 Þ 2 2

ð222Þ where yðzÞ ¼ 2ðz
zn Þ=ðznþ1
zn
1 Þ, zn are uniformly spaced, and similar equations hold for X0 , W, and W0 is found to provide the accuracy needed for tunneling calculations. a. Large Argument Case. On the face of it, Zi functions appear to offer no advantages over the Airy functions. The reason for introducing them is that ratios of the Zi functions (as shall appear during the matrix evaluations below) asymptotically approach the plane wave or exponential functions as the field vanishes, and so the methodology meshes well with the approach based on Eq. (196). Consider the case of c ¼
1, and examine the ratio Zið
1; zÞ=Zið
1; zo Þ where zo ¼ z(x ¼ 0). From the definition of Zi, to leading order 9 8 < 2i h i=



Ziði; zÞ 3=2 3=2 k2
k2o
sfx  exp
k2
k2o ; :3f Ziði; zo Þ ð223Þ h

1=2 i  exp
is k2
k2o x when k2o
k2  fx. The wave functions for non‐zero f are combinations of the Zi functions, or ck ðxÞ ¼ tðkÞZiðc; zðxÞÞ þ rðkÞZið
c; zðxÞÞ

ð224Þ

for c ¼ 1 (over the barrier) or i (under the barrier), and the choice of t and r for coefficients reflecting which of the terms (t in particular) correspond to an outgoing wave. Continuity of the wave function also requires the evaluation of the gradient, or ] 3 ] c ðxÞ ¼
f 1=3 sc2 z5=3 ck ðxÞ: ð225Þ ]x k 2 ]z

ELECTRON EMISSION PHYSICS

79

The quantity of interest embedded in Eq. (225) is the relation 2 31=3  ] f sc2  Ziðc; zÞ ¼ 4 5 ð4c
zÞFc ðzðzÞÞ
6z2 ]z Fc ðzðzÞÞ Ziðc; zÞ ]x 64z Fc ðzðzÞÞ  Diðc; zÞZiðc; zÞ

It follows from Eq. (220) that 0 11=3   f A Wc ðzðzÞÞ þ cW 0c ðzðzÞÞ Diðc; zÞ ¼ s@ zðzÞ   s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k2o
k2 þ fsx Wc ðzðzÞÞ þ cW 0c ðzðzÞÞ c

ð226Þ

ð227Þ

where, in the second line, z has been expressed in terms of the parameters introduced by Schro¨dinger’s equation. Eq. (227) is well behaved for the range of conditions characteristic of tunneling and therefore is of considerable use below. b. Small Argument Case. When z is small, numerical work benefits from expansions if library routines for the evaluation of the Airy functions are not available (Abramowitz and Stegun, 1965). In practice, truncating the series expansions such that six‐digit accuracy is preserved is sufficient for accurate numerical tunneling calculations. Therefore PðzÞ SðzÞ 0 1
0 1 AiðzÞ ¼ 2 1 32=3 G@ A 31=3 G@ A 3 3 ð228Þ PðzÞ SðzÞ 0 1þ 0 1 BiðzÞ ¼ 2 1 31=6 G@ A 3
1=6 G@ A 3 3 The truncated polynomials P(z) and S(z) are defined by PðzÞ ¼ 1 þ SðzÞ ¼





z3 285120 þ z3 9504 þ z3 132 þ z3 1710720





z 7076160 þ z3 589680 þ z3 14040 þ z3 156 þ z3 7076160

ð229Þ

80

KEVIN L. JENSEN

Gradients of the Airy functions are then evaluated by



z2 71280 þ z3 4752 þ z3 99 þ z3 142560



z 181440 þ z3 7560 þ z3 120 þ z3 S0 ðzÞ ¼ 1 þ 544320

P0 ðzÞ ¼

ð230Þ

Using Eqs. (228) – (230), small argument values of Di(c,z) can likewise be found. c. Wronskians of the Airy Functions. relation (Watson, 1995)

The Airy functions satisfy the

AiðzÞð]z BiðzÞÞ
ð]x AiðzÞÞBiðzÞ ¼

1 p

ð231Þ

for arbitrary z, a relationship useful due to Wronskians appearing for the matrix inverses below. In terms of the Zi functions, this becomes (where the gradient with respect to position is explicitly shown to avoid confusing primes to be gradients with respect to argument) Ziðc; zÞð]x Zið
c; zÞÞ
Zið
c; zÞð]x Ziðc; zÞÞ ¼

1 1=3 f sc 1
3c2 2p

ð232Þ

5. The Triangular Barrier The triangular barrier represents an example of the Airy function approach and is the basis for the derivation of the FN equation as originally given (Fowler and Nordheim, 1928) in the treatment of field emission. Consider the potential V(x) ¼ Vo – Fx. In the parlance introduced above, it becomes k2o
k2 þ sfx with s ¼
1. Matching the wave function and first derivative at x ¼ 0, corresponding to z(x¼0) ¼ zo, gives       t0 1 Ziðc; zo Þ Zið
c; zo Þ 1
1 ¼ : ð233Þ 0 0 r Zi ðc; zo Þ Zi ð
c; zo Þ r0 ik
ik For transmission over the barrier (c ¼ i), it follows that t0 ¼ t and r0 ¼ 0, but for under the barrier (c ¼ 1), the value of c changes to i when the wave emerges at the location defined by z ¼ 0. Consequently, a transition matrix must be introduced whenever c changes value. Therefore, for the case of under the barrier to over       Zið1; 0Þ Zið
1; 0Þ t0 t Ziði; 0Þ Zið
i; 0Þ ; ð234Þ ¼ Zi0 ð1; 0Þ Zi0 ð
1; 0Þ r0 0 Zi0 ði; 0Þ Zi0 ð
i; 0Þ

81

ELECTRON EMISSION PHYSICS

and for over the barrier to under     Ziði; 0Þ Zið
i; 0Þ t0 Zið1; 0Þ ¼ 0 0 Zi ði; 0Þ Zi ð
i; 0Þ r0 Zi0 ð1; 0Þ Use of the Wronskians shows that 8   t
i i > >   > < 1
1 0 t  0 ¼ 1i 0 t 1 r > > > : 2
i 1 0

Zið
1; 0Þ Zi0 ð
1; 0Þ

k2o > k2 k2o < k2

  t : 0

ð235Þ



ð236Þ

Now let us restrict attention to electron energies below the barrier maximum (under to over). Then the matrix equation to be solved is        1 Zið1; zo Þ Zið
1; zo Þ t 1
1
i i ¼ : ð237Þ r 0 ik
ik 1
1 Zi0 ð1; zo Þ Zi0 ð
1; zo Þ The solution for t(k) is revealed by expanding the matrices and finding tðkÞ ¼

2k : ðk
iDið
1; zo ÞÞZið
1; zo Þ
iðk
iDið1; zo ÞÞZið1; zo Þ

ð238Þ

The wave function becomes (outside the barrier to the right) ck ðxÞ ¼

2kZiði; zÞ : ðk
iDið
1; zo ÞÞZið
1; zo Þ
iðk
iDið1; zo ÞÞZið1; zo Þ

ð239Þ

Inside the barrier to the right, the Zi in the numerator would be replaced with Zi(1,z), but that case is ancillary to our present focus on the emitted current. If the electron energy is well below the barrier height, then   2ik Ziði; zÞ ck ðxÞ 
: ð240Þ ðk
iDið1; zo ÞÞ Zið1; zo Þ The employment of the Zi functions, argued to be useful when their ratios are taken, therefore reveals their utility. From the relation Ziði; zÞ ¼ Zið
i; zÞ, it follows that the transmission coefficient T(k) is (where the smaller terms neglected in Eq. 240 are kept)

 1=3  4k2 f 

 Tðk < ko Þ ¼ 2 2 2 2 2 1=3 pk 2 2 k þ Dið1; zo Þ Zið1; zo Þ þ p f k þ k þ Dið
1; zo Þ Zið
1; zo Þ

ð241Þ An analogous equation follows for emission over the barrier. It is a good pedagogical (if slightly pedantic and definitely tedious) exercise to examine the limit of a step function potential and demonstrate that, as expected, the

82

KEVIN L. JENSEN

wave function and transmission coefficient are as described previously in the derivation of Eq. (202). It is an exercise to show that moving from the Zi functions to the traditional Airy functions results in expression of Eq. (241) as Tðk < ko Þ ¼



4k2



k2 Aiðzo Þ2 þ Biðzo Þ2 þ p2 f 1=3 k þ f 2=3 Ai0 ðzo Þ2 þ Bi0 ðzo Þ2



 f 1=3 ; pk

ð242Þ where the center term in the denominator is a consequence of the Wronskian. Performing the same analysis for k > ko replaces zo in Eq. (242) by –zo, in contrast to Eq. (241), which instead becomes  1=3  4k2 f Tðk > ko Þ ¼ : ðk2 þ Diði; zo ÞDið
i; zo ÞÞZiði; zo ÞZið
i; zo Þ þ p2 f 1=3 k pk ð243Þ The FN equation, developed for field emission from metals, was principally concerned with electron energies below the barrier maximum for large work functions. Therefore, the asymptotic limit of Eq. (241) is desired for zo ¼ jk2o
k2 j3=2 =f 2=3  k3 =f 2=3  1. Let twice the AUC term be designated by 4k3 =3f 2=3 ¼ y, and use the asymptotic expansions for the under‐the‐barrier Zi 8 9 < y= f 1=6 Zið1; zo Þ  pffiffiffiffiffiffi ð3  1Þexp  : 2; 4 pk ð244Þ

 k2 þ Dið1; zo Þ2 Zið1; zo Þ2  k2o Zið1; zo Þ2 The inclusion of a factor of (2) in the definition of y is a slight departure from other analyses (Jensen, 2001) where y is identified with the AUC term directly. Eq. (241) becomes Tðk < ko Þ ¼

16kk : k2o f4ey þ e
y g þ 8kk

The FN approximation to the transmission coefficient is then  4k 4 TFN ðkÞ ¼ lim TðkÞ ¼ exp
k3 : ko !1 ko 3f

ð245Þ

ð246Þ

Even though the potential is sharply peaked, it is clear that the coefficient is field independent and the argument of the exponential is the AUC term, both keeping in line with Eq. (190).

83

ELECTRON EMISSION PHYSICS

For k > ko, an analysis analogous to the one leading to Eq. (245) in turn gives rise to the asymptotic approximation Tðk > ko Þ ¼

4kk ðk þ kÞ2

:

ð247Þ

The two limits of Eq. (245) and (247) are suggestively similar, but not quite the same. They suffer from the problem that both vanish when k ¼ ko (i.e., k ¼ 0), whereas neither Eq. (241) nor Eq. (243) vanishes. Pursuing an expansion that is correct through the point k ¼ ko may appear churlish, but the effort belies a subtlety that is useful for the analysis of other barriers, in particular the quadratic barrier considered below. A careful analysis shows that the problems at k ¼ko arise from the presence of z1/4 in the denominator of the asymptotic (large z) expansion of Zi(c,z) in Eq. (210). The simplest approximation is to remove

1=8 the singularity by appending a small, finite term, as in 1=4 zo ! z2o þ p2 . The same analysis that yielded Eq. (247) then gives Tðk > ko Þ ¼

4kðk4 þ f 4=3 p2 Þ

1=4

k2 þ k2 þ 2kðk4 þ f 4=3 p2 Þ

1=4

:

ð248Þ

The value of p is found by demanding that Eq. (248) be valid at k ¼ ko, using Eq. (242), and the zero‐argument terms given in Eq. (228). The resulting expression depends on both ko and f and it can be shown that 0 12 2 3
2 4=3 2=3 2=3 2 2 9k 3 f 3 k o 5 p ¼ @ oA 4 þ 4p Gð1=3Þ2 Gð2=3Þ2 ð249Þ 0:398593k4o ¼

2 k2o þ 0:531457f 2=3 for vanishing field, p approaches a barrier‐independent constant. How does this help? First, note that the term e
y is negligible in Eq. (245) except near y ¼ 0, and so neglecting it in general is useful. Second, as T(k) is a continuous function of k, then y(k) should be likewise continuous. As y depends on k2, this amounts to continuity in E. We therefore take y to be the AUC factor for energies below the barrier maximum but to be the linear continuation in E of that function for energies above the barrier maximum. Consequently, the procedure is to replace y for energies above the barrier maximum with the linear extension y0 ðm þ FÞðE
m
FÞ, where the prime on y denotes derivative with respect to argument, when the energy exceeds the barrier maximum. For the FN triangular barrier, such a procedure is trivial: y0 ðEÞ ¼ ð2=
hF Þ½2mðm þ F
E Þ1=2 vanishes at the barrier maximum, and so y vanishes for energies above the barrier. In contrast, for barriers

84

KEVIN L. JENSEN

where y0 does not vanish at the barrier maximum, the prescription is to linearly extend the below‐barrier results to above the barrier. For the triangular barrier, then, the form of T(k) valid for all k is

T ðkÞ ¼

16kðk4 þ f 4=3 p2 Þ

1=4

4ðk2 þ k2 Þexp½yðkÞ þ 8kðk4 þ f 4=3 p2 Þ

1=4

;

ð250Þ

where yðkÞ ¼ ½4k3 =3f , for k < ko and 0 for k ko, and where k2 ¼ jk2o
k2 j. Observe that in the limit of vanishing field, Eq. (202) is recovered. Consider the performance of Eq. (250) for copperlike parameters, that is, ˚ , for which m ¼ 7.0 eV and F ¼ 4.6 eV and an applied field of 0.4 eV/A ˚ and f ¼ 0.104987 1/A ˚ 3. Figure 23 compares the exact result ko ¼ 1.76366 1/A with the FN approximation [Eq. (246)] and Eq. (250). The p prescription of Eq. (246) works quite well. In addition, the exact solution is shown for several fields in Figure 24. The pedagogical value of Eq. (250) is sufficient to justify the effort invested in its derivation revealing the nature of the denominator; loosely, the transmission coefficient is approximately of the form Taprx ðkÞ 

ð251Þ

Exact Analytic FN m&m+Φ

1.0 Transmission coefficient

C ðk Þ ; 1 þ expðyðkÞÞ

0.8 0.6

m = 7 eV Φ = 4.6 eV F = 0.4 eV/Ang m ko = 1.745 Ang−1

0.4 0.2 0

7

8

9

10 11 Energy [eV]

m+Φ

12

13

14

FIGURE 23. Comparison of the numerically evaluated transmission probability using the Zi functions [exact Eq. (241)] with the traditional Fowler Nordheim equation [FN Eq. (246)] and the analytical approximation Eq. (250).

85

ELECTRON EMISSION PHYSICS

Transmission coefficient

1.0

m = 7 eV Φ = 4.6 eV

0.8

2 eV/nm 4 eV/nm 6 eV/nm 8 eV/nm 10 eV/nm µ m +Φ

0.6 0.4 0.2 0.0

5

10

15

20

Energy [eV] FIGURE 24. The triangular barrier emission probability calculated according to Eq. (250) for copper-like parameters for various fields.

where C(k) for large k approaches unity. The form of Eq. (251) is a general form that we wish to retain below. Above the barrier in general, yðk > ko Þ can be approximated by 0

yðk > ko Þ ¼ yo ðE ðko ÞÞ þ yo ðE ðko ÞÞðE ðkÞ
E ðko ÞÞ;

ð252Þ

where the prime indicates derivative with respect to energy, even though in the case of the triangular barrier, both y and its first derivative vanish at E ¼ m þ F (that is, saying the AUC factor vanishes above the barrier is a consequence of the special dependence yðE Þ / ðm þ F
E Þ3=2 for E < m þ F for the triangular barrier). In general, y(E) and its first derivative do not so vanish (such as for quadratic barriers), and therefore the form of Eq. (252) is useful in a relation such as Eq. (251).

C. Wentzel–Kramers–Brillouin WKB Area Under the Curve Models 1. The Quadratic Barrier The quadratic barrier can be generally written for |x| < L as  Vquad ðxÞ ¼ Vo 1


x 2 L

:

ð253Þ

86

KEVIN L. JENSEN

The AUC expression for y then is simple to evaluate and yields vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 0 12 9 sffiffiffiffiffiffiffið xo u u < u 2m x = tVo 1
@ A yquad ðE Þ ¼ 2
Edx 2 : L ; h
xo
0 1 E ¼ pko L@1
A Vo

ð254Þ

where ð
hko Þ2 =2m ¼ Vo and xo ðE Þ ¼ L½1
ðE =Vo Þ1=2 . The extension of yquad ðE > Vo Þ is then trivial as yquad ðE < Vo Þ is already linear. Consequently, the approximation to the quadratic barrier using the form suggested by Eq. (251) is then   
1 : ð255Þ Tquad ðE Þ  1 þ exp yquad ðE Þ The performance of Eq. (255) is shown in Figure 25 for copperlike parameters (m ¼ 7.0 eV, F ¼ 4.6 eV) and a barrier width of 2L ¼ 1 nm . Clearly, the performance near the barrier maximum (11.6 eV) is quite good; less clearly visible is that near E ¼ m, Eq. (255) is approximately 23% larger than the Airy function solution. Before much is made of the latter discrepancy, recall that in light of the generally unknown surface conditions, there are substantial differences between real surfaces and models that purport to describe them. But before much is made of that, the absence of a perfect model is not license to use a maladapted one. Models such as Eq. (255) prove their utility when the emission current contains contributions near the barrier maximum, as in thermionic and photoemission, a point returned to in the following text.

Transmission coeff.

1.0

Exact exp(−θ) 1/[1 + exp(θ)]

0.8 0.6

2L = 1 nm Copper-like: m = 7.0 eV Φ = 4.6 eV

0.4 0.2 0.0

8

9

10

11 12 13 14 Energy [eV] FIGURE 25. Comparison of the exact quadratic barrier transmission probability with the standard area under the curve approximation exp(
y) and Eq. (255).

87

ELECTRON EMISSION PHYSICS

2. The Image Charge Barrier The last of the analytic models to be considered is arguably the most influential one, as it is the basis for the thermionic (Richardson) and field emission Fowler Nordheim (FN) equations treated below. Consider, therefore, the potential given by the image charge potential Eq. (110), for which the associated AUC expression is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1ffi ð xþ ðEÞ u u Q u2m @ t 2 m þ F
Fx

E Adx yimage ðEÞ ¼ 2 x h
x
ðEÞ ð256Þ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 ðm þ F
E Þ  ðm þ F
E Þ
4FQ x ðEÞ ¼ 2F Introducing a change of variables governed by the length L(E) ¼ xþ – x, Eq. (256) becomes 4L pffiffiffiffiffiffiffiffiffiffiffiffiffi x
 2mLF R0 yð E Þ ¼ ; ð257Þ h
L where R0 ðxÞ ¼

ð p=2 0

cos2 ðsÞsin2 ðsÞ  1=2 ds: x þ sin2 ðsÞ

ð258Þ

The form of Eq. (257) has a certain utility to it: it is in the form of a product of a length term (L) with a wave number term related to the height of the barrier above the Fermi level (F ¼ FL) with a dimensionless correction term (R0(x)) accounting for the difference between the image charge barrier and the triangular barrier, a feature repeated for other potentials. Two limits of R0(x) are easily found to be R0 ð0Þ ¼ ð1=3Þ and R0 ðx  1Þ  p=ð16x1=2 Þ. The FN triangular barrier result is obtained by setting Q ¼ 0 and using R0(0). A more detailed analysis based on a partial summation of the series involved provides better approximations (Jensen, 2001), namely, 8 pffiffiffiffiffiffiffiffiffiffiffi 0 2 31 0 1 > > 1 þ x x > @1408 þ px4336ln@ A
1515A > > > > 1þx 4224 > > > < R0 ðxÞ  0:35657
0:28052pffiffixffi þ 0:086441x > > > > pffiffiffi > > > p x > > > : 4ð4x þ 1Þ

ðx < 0:125Þ ð0:125  x  1:0Þ ; ðx > 1:0Þ

ð259Þ

88

KEVIN L. JENSEN

R0(x) Small x Large x Mid x Domains

R0(x)

0.3

0.2

0.1

0

0

0.5

1 Sqrt(x)

1.5

2

FIGURE 26. Comparison of the exact Eq. (258) with its approximation Eq. (259).

where a critical feature, namely the logarithmic dependence on the small x behavior, is shown to exist. The performance of Eq. (259) compared to Eq. (258) is shown in Figure 26: before much is made of it and its accuracy, the utility of equations such as Eq. (259) have been permanently eclipsed by an approximation due to Forbes (2006), discussed in greater detail below, rendering further discourse on Eq. (259) a bit anachronistic and only of historical interest. a. Expansion of y near E ¼ m. What passes for the ‘‘traditional treatments’’ of the FN equation often is based on the formulation of Murphy and Good (1956) to account for image charge modifications on the current density formulas through the introduction of functions v(y) and t(y) (see Forbes and Jensen, 2001, for tabulated values), which arise when Eq. (257) is rendered linear in energy E about the expansion point m. They are related to the R0(x) for 0  y  1 functions by 0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
sin ð xÞ A vðcosðxÞÞ ¼ 3 sin3 ðxÞR0 @ 2sinðxÞ 2 3 ð260Þ 2 tðcosðxÞÞ ¼ 41 þ cotðxÞ]x 5vðcosðxÞÞ 3

The traditional form tðyÞ ¼ 1
ð2=3Þy]y vðyÞ is more often encountered (Modinos, 1984). The literature is replete with clever approximations to accomplish various ends, although the most common end sought is to approximate effective emission area, work function, or both from current versus voltage data rendered on an FN plot (Forbes, 1999a). Although the Forbes approximation to v(y) is deferred to later, in the literature much effort is often devoted to the

ELECTRON EMISSION PHYSICS

89

form vðyÞ  vo
y2 so that a plot of current density versus field on an FN plot is explicitly linear, and so there is historical interest in describing such efforts. Expanding v(y) about yo to order y2 results in vquad ðyÞ  vðyo Þ þ ðy2
y2o Þ

3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2zo þ 1
3Rðzo Þ þ ð2zo þ 1ÞR o ðzo Þ ; 4 ð261Þ

where zo is the argument of R0 evaluated at y ¼ yo. If yo is chosen such that the coefficient of y2 is identically unity, then yo ¼ 0.599161, and vquad ðyÞ  0:936814
y2 :

ð262Þ

A widely used form vðyÞ  vo
y2 , with vo ¼ 0.95, was introduced by Spindt et al. (1976). It is a challenging but ultimately pointless exercise to aspire to an analog of Eq. (262) for t(y); often it is merely approximated by a constant. The fact is that low‐order Taylor expansions perform poorly as a consequence of the embedded logarithmic dependence hinted at in Eq. (259). A crude three‐point fit is 0 1 1 tðyÞ  ð1
yÞð1
2yÞtð0Þ þ 4yð1
yÞt@ A
yð1
2yÞtð1Þ ð263Þ 2 ¼ 1 þ 0:06489y þ 0:0458308y2 pffiffiffi where t(0) ¼ 1, t(1/2) ¼ 1.0439, and t(1) ¼p= 8 and is compared to exact values (Figure 27). This, however, is only a temporary mathematical ‘‘fix’’; better approximations are described in the discussion of the Forbes approximation; fortunately, that approximation is worth the wait. Regardless of how v(y) and t(y) are obtained, the linearized y(E) is given by (where the subscript ‘‘fn’’ refers to Fowler–Nordheim, as this form is needed in the derivation of the FN equation, as given by Murphy and Good): bfn þ cfn ðm
E Þ F 0 1 pffiffiffiffiffiffiffiffiffiffi 4 pffiffiffiffiffiffiffiffiffiffiffiffi3 @ 4QF A bfn ðF Þ ¼ 2mF v 3
h F 0 1 pffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffi @ 4QF A 2mF t cfn ðF Þ ¼ hF
F

yfn ðE  mÞ ¼

ð264Þ

90

KEVIN L. JENSEN

1.0

1.12

0.8

1.10

v(y) vquad(y) t(y) t quad(y)

1.08 1.06

t(y)

v(y)

0.6 0.4

1.04 0.2 0.0

1.02 0

0.2

0.4

0.6

0.8

1

1.00

y FIGURE 27. Performance of the “crude” quadratic approximations to v(y) and t(y) [Eqs. (262) and (263), respectively] compared to exact (numerically evaluated) results.

where the unusual choice of bfn/F is made so that when the quadratic form of v(y) is used, the resulting intercept is linear in F—a useful feature in the representation of current density on an FN plot of lnðJ =F 2 Þ versus 1/F. In the limit that Q approaches 0 (i.e., as the image charge is neglected and the potential barrier becomes triangular), the original FN representation, which would be obtained from the linearization in energy about m of the argument of the exponent in Eq. (246) [and as suggested in Eq. (252)] is recovered. b. Expansion of y near E ¼ m þ f: The Quadratic Barrier. When the work function is low, or when the temperature is high, the transition from tunneling (under the barrier) to thermal (over the barrier) emission occurs (Gadzuk and Plummer, 1971) and there the expansion point needs to be taken, not at the Fermi level, but closer to the potential maximum. Near the barrier maximum, the image charge potential resembles an inverted parabola, that is, a quadratic potential, in which case the linear expansion is simply Eq. (254) but with Vo and symmetry axis of the quadratic potential dictated by the image charge parameters Vo ¼ Vimage(xo) and ]2x Vimage ðxo Þ ¼ ]2x Vquad ðxo Þ, (the correspondence is shown for several fields in Figure 28), resulting in rffiffiffiffiffiffiffi o pffiffiffiffiffiffiffiffiffiffi p 2mn F
4QF þ ðm
E Þ : ð265Þ yquad ðE Þ ¼ 2
h F The behavior of Eq. (265) is shown for copper parameters at fields characteristic of tunneling in Figure 29, labeled by the acronym SICT (standard image charge theta) and compared to AICT (approximate image charge

ELECTRON EMISSION PHYSICS

91

11 1 eV/nm

Potential [eV]

10

9

8

7

5 eV/nm Image Quadratic 9 eV/nm

Cu µ = 7.0 eV Φ = 4.6 eV

3

6 9 12 15 Position [angstroms] FIGURE 28. Comparison of the image charge potential (thick line) to the quadratic barrier potential (thin line) accurate near the apex for increasing fields.

theta) designating Eq. (264) and Quad designating Eq. (265). The figure showing the ratios of the approximations with the WKB y shows that AICT performs well near the Fermi level, but Eq. (265) is accurate near the barrier maximum, where the image charge potential is better represented by a quadratic, and also better for high fields, where the triangular nature of the barrier is suppressed. c. Reflection Above the Barrier Maximum. Use of Eq. (265) for energies larger than the barrier maximum as per the prescription of Eq. (252) worked well for the quadratic barrier. For the image charge potential, however, the correspondence is not quite as cozy: evaluations of T(E) using numerical methods (such as those described below) show that T(E) does not approach unity for E > m þ F nearly as rapidly as the linear extension of y model suggests. Another factor contributes, as suggested by the differences between the triangular barrier and quadratic barrier models: C ðEðkÞÞ for the former, as inferred from a comparison of Eqs. (250) and (251), is a nontrivial creature, whereas for the latter, as inferred from the success of Eq. (255), it is unity, an effect therefore inferred to be related to the abruptness of the triangular barrier compared to the far more composed rise of the quadratic barrier. The image charge barrier has elements of both—an abrupt rise near the origin due to the image charge term, and a leisurely decline far from the origin due to the field term. A good analytic model does not present itself, but the Bohm analysis suggests a reasonable kludge (Jensen, 2003b). Consider an incident plane wave to the left of the first zero of the image charge potential barrier. To the right, let the wave function be approximated by ck ðxÞ ¼ tðkÞRk ðxÞexpðiSk ðxÞÞ, but after x ¼ xþ assume that the potential

92

KEVIN L. JENSEN

(a) 60

SICT1 AICT1 Quad1

50 1 V/nm

q (E)

40 30 20 10 0

5

6

7

8 9 Energy [eV]

10

11

(b) 10 SICT 9 AICT 9 Quad 9 SICT 5 AICT 5 Quad 5

8 5 V/nm

q (E)

6 4 2

9 V/nm

0 −2

5

6

7

8 9 Energy [eV]

10

11

0.9

m 0.6

6

7

8 9 Energy [eV]

1 eV/nm

0.7

5 eV/nm

0.8

9 eV/nm

Ratio {qaprx (E)/q(E)}

(c) 1.0

10

11

FIGURE 29. (a) Comparison of the linear expansions for the y function evaluated using the approximate image charge y (AICT) and the quadratic approximation (Quad), compared to the exact result (SICT) for a field of 1 GV/m. (b) Same as (a), but for the higher fields of 5 GV/m and 9 GV/m. (c) The approximations (AICT and Quad) to y compared to the numerical evaluation. As expected, the approximations are good only near the expansion points of the chemical potential (i.e., Fermi level).

93

ELECTRON EMISSION PHYSICS

12

Potential [eV]

9

6 Copper-like m = 7.0 eV Φ = 4.6 eV L = 1.0 nm

3

0 0

2 4 6 8 Position [angstroms]

10

FIGURE 30. Quadratic potential (inverted parabola) for copper-like parameters.

is flat such that the wave function resumes its plane wave behavior (an example for copperlike parameters being given in Figure 30). Equating wave function and first derivative at x ¼ x– for energies in excess of the barrier maximum suggests that t(k) is given by tðkÞ ¼

2ikexpðikx

iS Þ ikR þ ]x R þ iR]x S

jtðkÞj2 ¼ n

4k2 ð]x RÞ2 þ R2 ðk þ ]x S Þ2

o

ð266Þ

where x is evaluated at x– (for simplicity the k subscript (k)on R and S is suppressed) and EðkÞ ¼
h2 k2 =2m. Neglecting the quantum potential indicates qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 that ]x S  k2
kv ðxÞ and R  ½E=ðE
V ðxÞÞ1=4 , where kv is defined by V ðxÞ ¼
h2 kv ðxÞ2 =2m. It follows that S(x–) ¼ 0 and R(x–) ¼ 1 because V(x–) is identically 0 by definition, and, in this approximation, "  # F xo 2 ]x Rðx
Þ ¼
1 : ð267Þ 4EðkÞ x
Joining components suggests that C(k) for the image charge potential is given by |t(k)|2, or E3 ð268Þ Cimage ðEðkÞÞ ¼   2 : 2 2 2 x h
F o E 3 þ 128m x

1

94

KEVIN L. JENSEN

Transmission probability

1.0 0.8 0.6 T(E) hyp-SICT AICT Analytic

0.4 0.2 0

6

9 12 Energy [eV]

15

FIGURE 31. Product (thick line) of the energy-dependent coefficient in Eq. (268) to the hyperbolic tangent approximation (thin line) to exact results. Also shown is the AICT approximation for comparison, which performs disastrously near the barrier maximum.

The impact of Eq. (268) on the tanh‐WKB model is shown for copperlike parameters under an applied field of 8 eV/nm for the image charge potential in Figure 31. At the barrier maximum (E ¼ m þ F ¼ 8.21 eV), C(k) is approximately 88.4% and slowly increases to 95% at E ¼ m þ F ¼ 11.6 eV. The improvement is evident. The dependence of C on energy is nevertheless comparatively weak compared to e
yðE Þ , and therefore, it is often enough to replace C(E) by C(Em), where Em is the location of the integrand maximum (which, for thermionic emission, is approximately at m þ f) when the current density is being evaluated. D. Numerical Methods The numerical evaluation of T(k) uses the modified Airy function approach, replacing the plane waves previously considered in Eq. (197). Analogously, at each region where a change in slope or a discontinuity in height occurs (or both), the relation between the coefficients to the left and right is tn
1 rn
1

!

¼




1=3

2p

fn
1 sn
1 cn
1 3c2n
1
1 Ziðcn ; zn Þ ]x Ziðcn ; zn Þ



]x Zið
cn
1 ; zn
1 Þ


Zið
cn
1 ; zn
1 Þ


]x Ziðcn
1 ; zn
1 Þ

Ziðcn
1 ; zn
1 Þ

Ziðcn ; zn Þ ]x Zið
cn ; zn Þ

!

tn

!

!

ð269Þ

rn

where the coefficient is a consequence of the Wronskian of the Zi function. As in the treatment of the triangular barrier, if the nth region includes a transition from above the barrier to below (or vice versa), then the transition

95

ELECTRON EMISSION PHYSICS

matrices of Eq. (236) are required. In all other respects, the methodology is analogous to the triangular barrier and square barrier examples—albeit with more segments, necessitating greater attention to when the transition matrices must be invoked. (The methodology here is analogous to, but simpler, than that found in Jensen, 2003b.) Consider the quadratic and image charge potentials and their numerical solution as case examples. 1. Numerical Treatment of Quadratic Potential A discretization of the quadratic potential using 24 linear segments is shown in Figure 30 where copperlike parameters are used. From this potential, the transmission coefficient was calculated for 200 values of energy (Figure 32). The tanh‐WKB approximation 1=ð1 þ eyðE Þ Þ is compared to the numerical T(E), and the more familiar WKB approximation e
yðE Þ , where y is as given in Eq. (254). On this scale, the tanh‐WKB approximation works well. The numerical calculation, from the generation of the potential, the initialization of the Zi functions, the evaluation of T(E) for 200 cases, and the output of the data is rapid, taking less than a second on a contemporary desktop computer. 2. Numerical Treatment of Image Charge Potential Representing the image charge potential as a sequence of piece‐wise linear regions is more art than science: where the potential varies rapidly and nonlinearly (near the origin), many small potential regions are required, whereas with a predominantly linear potential (far away where the image charge term is negligible), the length of the segments can be substantially

Transmission probability

1.0 Numerical tanh–WKB

0.8

WKB 0.6 0.4 0.2

Quadratic potential

0 6

8

10 Energy [eV]

Copper-like m = 7.0 eV Φ = 4.6 eV L = 1.0 nm 12

14

FIGURE 32. Numerical solution of the quadratic barrier of Figure 30 compared to the tanh-approximation and the exp(-y) approximation.

96

KEVIN L. JENSEN

longer. Generally, precision of a method does not necessarily guarantee the accuracy of its result. The relation between art and precision takes some quantification. Consider the representation of image charge potential, starting with a small number of segments and increasing their number. A crude measure of ‘‘accuracy’’ is whether doubling the number of segments results in a negligible change in the variation of the transmission probability. In particular, consider the following schemes referred to as ‘‘linear,’’ ‘‘quadratic,’’ and ‘‘proper,’’ in which x(i) is evenly spaced, the length of the segments increases quadratically, and the length of the segments reflects the importance of the region, respectively. They are 0 1 i
1 A xlinear ðiÞ ¼ x
þ ðxþ
x
Þ@ N
1 0 12 i
1 A xquad ðiÞ ¼ x
þ ðxþ
x
Þ@ N
1 8 0 13=2 ð270Þ > > > N
2i > A > 2i  N x
þ ðxo
x
Þ@ > > < N
2 xproper ðiÞ ¼ 0 12 > > > 2i
N A > > x þ ð x þ
xo Þ @ 2i > N > > : o N where xþ, x–, and xo are the larger and smaller zeros of V(x) and the location of the barrier maximum, respectively. The schemes can be characterized as follows. Linear takes no account of details of the potential, and is therefore expected to perform poorly. Quad accounts for the steep variation near the origin and minimizes the variation far away, but it does not take notice of the actual barrier maximum location and value, and in particular, has coarser discretization there than near x–. Proper takes pains to discretize finely near the barrier maximum and less so as the points move further from xo (it is proper only in the sense that it respects details of the potential, not that it is the optimal choice, which it is not), and—significantly—the maximum of the potential is one of the grid points. If the calculation of T(E) by numerical means is accurate, then doubling the number of linear segments in the modified Airy function approach has minimal impact. The change in doubling the number of points chosen for the potential as per Eq. (270) is shown in Figure 33. As a consequence of the increase in the number of segments, the effects on the numerically determined T(E) are shown in Figure 34. Clearly, the linear method is pathetic: a large portion of the potential is shaved off in the N ¼ 8 case, which, by intuition

97

ELECTRON EMISSION PHYSICS

Potential energy [eV]

(a) 10 Image potential N=8 N = 16

8 6

Linear x(i) 4

Copper-like m = 7.0 eV Φ = 4.6 eV F = 5 eV/nm

2 0 0

4

8 12 16 Position [angstroms]

20

24

(b) 10 Image potential N=8 N = 16

8 6 4 2 0 0

4

8

12

16

20

24

Potential energy [eV]

(c) 10 Image potential N=8 N = 16

8 6

Proper x(i) 4

Copper-like m = 7.0 eV Φ = 4.6 eV F = 5 eV/nm

2 0 0

4

8 12 16 Position [angstroms]

20

24

FIGURE 33. Discrete representation of the image charge potential (a) linearly spaced regions; (b) quadratically spaced regions, (c) fine spacing near maximum, coarse in linear regions.

98

KEVIN L. JENSEN

Transmission probability

(a) 1.0 Copper-like m = 7.0 eV Φ = 4.6 eV F = 5 eV/nm

0.8 0.6

N=8 N = 16

0.4 0.2 Linear x(i) 0 7

8

9 10 Energy [eV]

11

Transmission probability

(b) 1.0 Copper-like m = 7.0 eV Φ = 4.6 eV F = 5 eV/nm

0.8 0.6

N=8 N = 16

0.4 0.2 Quadratic x(i) 0 7

8

9

10

11

Energy [eV]

Transmission probability

(c) 1.0 Copper-like m = 7.0 eV Φ = 4.6 eV F = 5 eV/nm

0.8 0.6

N=8 N = 16

0.4 0.2 Proper x(i) 0 7

8

9

10

11

Energy [eV] FIGURE 34. (a) Comparison of the N = 8 and N = 16 linear schemes: agreement is poor, therefore accuracy is poor. (b) Comparison of the N = 8 and N = 16 quadratic schemes: agreement is moderate, therefore accuracy is moderate. (c) Comparison of the N = 8 and N = 16 proper schemes: agreement is close, therefore accuracy is good.

ELECTRON EMISSION PHYSICS

99

born of the WKB AUC method, will have predictable consequences—namely, T(E) is shifted to lower energies. The quadratic method fares better, but changes in the AUC factors near the barrier maximum have a noticeable impact. Finally, in the proper method, the scheme was designed with a goal of minimizing the discrepancies between the AUC factors; indeed, N ¼ 24 does not result in a readily discernible change in the behavior of T(E) compared to N ¼ 16. Accuracy, in a numerically cognizant interpretation, therefore implies rapid convergence of TN(E) as the parameter N increases. 3. Resonant Tunneling: A Numerical Example The investment behind the Airy function approach was considerably more significant than the AUC approaches based on the WKB method, and yet the methods generally yield comparable results for tunneling sufficiently below the barrier maximum. Given that surface conditions are extraordinarily complex (Haas and Thomas, 1968; Mo¨nch, 1995; Prutton, 1994), or that surface roughness itself (much less a deliberately pointed cathode geometry, such as Spindt‐type or carbon nanotube field emitters) introduces complications that cause the macroscopic applied field to differ substantially from the field at the emission site, effort directed toward the accurate calculation of the transmission probability seems to be the obsession of the aesthete. There are two responses. As a general matter, the presence of unknowns or impenetrable complexity is not license for indolence. As in philosophy, knowing what is not the case bounds what is, thus making even simple models inordinately useful. As a matter of practical importance, the AUC fails spectacularly if resonance contributes to the tunneling current; consider the treatment of an adsorbate on the surface of a metal as discussed in the magisterial tome by Gadzuk and Plummer (1973) and its more recent incarnations (Binh et al., 1992), which motivate (but are not identical to) an example considered below. A systematic treatment of resonance has been dealt with elsewhere (Jensen, 2003a,b) and is characterized by numerical gymnastics. Instead, a pleasantly straightforward numerical model that captures the main points will dominate the present focus. Returning to the FN triangular barrier, consider the excision of a small rectangular region from the potential barrier as shown in Figure 35. The potential is characterized by four grid points and one field, making for a particularly simple application of the matrix method. In form, the structure is similar to that of an RTD (Tsu and Esaki, 1973), (an account of the development of the idea behind resonant tunneling and its relationship to the FN equation is in the Nobel lecture of Esaki, 1973), the theoretical analysis and simulation of which has received considerable attention (Frensley, 1990; Price, 1988), except that electrons are not incident from the left, and metallic

100

KEVIN L. JENSEN

V(x) Grid Points ion

Potential [eV]

10

5 Barrier = 11.6 eV Field = 4 eV/nm

0

Well = −13.6 eV Width = 0.4 nm

−5 0

5

10 15 Position [angstroms]

20

25

FIGURE 35. A linear segment potential (linear black lines with gray dots at segment origins) containing a well for which a resonant state will occur, constructed to mimic the potential of an ion (shown in dark gray) but with minimum complexity.

parameters shall be considered (which sidesteps effective mass variation problems). Such a model is suggestive of (albeit in a simplistic fashion), for example, field emission from a single atom tip (Binh et al., 1992), the potential introduced by a barium atom on tungsten in a dispenser cathode (Hemstreet, Chubb, and Pickett, 1989), and defects at metal‐semiconductor interfaces (Mo¨nch, 1995). Therefore, the blue line marked ‘‘ion’’ shows the Coulomb potential associated with a screened charge outside the surface or o Vion ðxÞ ¼
Vion expð
ajx
xion jÞ=jx
xion j. The parameters of the example potential are again copperlike (Vo ¼ m þ F ¼ 11.6 eV, F ¼ 4 eV/nm), the well region is 0.4 nm wide and 13.6 eV deep (the ‘‘ion’’ curve is obtained from Voion ¼ 13.6 eV, xion ¼ 0.7 nm, and a ¼ 2 nm
1). The numerically calculated transmission probability is shown in Figure 36, along with the transmission probability for the FN triangular barrier, which is orders of magnitude smaller. Three resonant levels manifest themselves as peaks in ln(T(E)), their locations given coarsely by the infinite well energy levels. The triangular barrier with a square well excised is a convenient choice, because the transmission probability can be envisioned as a consequence of a big triangular barrier of height Vo with a smaller triangular barrier of height Vo – FL (where L is the location of the LHS of the well) excised from it, and an even smaller triangular barrier of height Vo – F(LþW) (where W is the width of the well) reinserted, inasmuch as the AUC formulation is only concerned about the barrier characteristics for the area above the energy E. Thus, it appears as though the transmission probability T(E) can be approximated as

101

ELECTRON EMISSION PHYSICS

0

ln {T(E)}

−10 −20 T(E) TFN

−30

Tbarr −40 2

4

6

8

10

12

Energy [eV] FIGURE 36. The transmission probability of Figure 35: TFN is the Fowler–Nordheim transmission probability without the excised well region; see Eq. (271) for the definition of Tbarr.

 T ðE Þ  TFN ðVo ; E Þ

TFN ðVo
F ðL þ W Þ; E Þ ; TFN ðVo
FL; E Þ

ð271Þ

where the term in the curly brackets f. . .g is referred to as Tbarr in Figure 36. Dividing T(E) by the RHS of Eq. (271) (i.e., we assume T ðE Þ  TFN Tbarr Tres and isolate Tres) therefore reveals the resonances Tres in stark contrast, as shown in Figure 37. Resonances are often represented as Lorentzians of the form (Price, 1988) To Tres ðE Þ  p

( )
1  E
Eres 2 þ1 ; d

ð272Þ

where the factor of p anticipates that, for small d, a Lorentzian behaves analogously to a Dirac delta function when integrated with other smoothly varying functions. Aside from the resonances, the transmission probability is handled quite well by products and ratios of AUC terms in that away from the resonances, the ratio is of order unity. If the resonances are indexed from n ¼ 1 to 3 for lowest to highest, then d(n) ¼ 0.006115, 0.061156, 0.470128; Eres(n) ¼ 0.87291 eV, 6.8533 eV, and 11.954 eV; and To(n) ¼ 384,085.00, 2520.58, and 6.32495, respectively. The width of the energy spread coupled with the magnitude of the coefficient guarantee that the presence of resonances will cause a substantial increase in the transmitted current. Therefore, when resonances are possible, near the energy levels of the well, AUC‐like formalisms miss the physics: greater diligence is demanded, and the Airy function approach provides it.

102

KEVIN L. JENSEN

12 ln {Transmission prob.}

10

T(E)/(Tbarr x TFN)

n=1

Σ lorentzians

8 6

n=2

4 2

n=3

0 −2 −4

2

4

6

8

10

12

Energy [eV] FIGURE 37. The extraction of Tres from T(E), compared to the sum over Lorentians modeling the resonances.

E. The Thermal and Field Emission Equation In addition to the transmission probability, the distribution of electrons in energy is needed for the estimation of current density. The emitted distribution depends on the particulars of the barrier, whether the majority of electrons tunnel through or are emitted over the barrier—or some combination thereof. To evaluate the total current density, a naive approach to evaluate the current density for a given momentum k, shown previously to be jtrans ¼ jtðkÞj2
hk=m, is simply to integrate jtrans(k) with its distribution function fo(k) (the problems with this approach have been examined in the discussion of the Wigner function representation for current density). The Tsu–Esaki formula (Tsu and Esaki, 1973) for current differences between left‐ and right‐flowing electrons from opposite boundaries uses such a method; letting dE=
h ¼ ð
hkdk=mÞ and T(E) ¼ |t(k)|2, ð   q TðEÞ foleft ðE Þ
foright ðE Þ dE: J¼ ð273Þ 2p
h For supply functions based on the FD distribution (as for RTD simulations) it follows that   ð qm 1 þ ebðm
EÞ TðEÞln J¼ dE; ð274Þ 1 þ ebðm
E
’Þ pb
h2

ELECTRON EMISSION PHYSICS

103

where ’ is the bias drop across the RTD structure. For the equations of electron emission, however, foright ¼ 0; coupled with the general transmission probability given by Eq. (251) and taking y(E) to be linear in E suggests that ð  


1 qm 1 JðF ; TÞ ¼ CðEÞln 1 þ ebF ðm
EÞ 1 þ ebT ðEo
E Þ dE; ð275Þ 2 pbT
h 0 where bT ¼ 1/kBT and bF are the slope factors of the supply function and transmission coefficient, respectively, in units of inverse energy, the notation serving to emphasize their analogous role. Eq. (275) is the general form from which limiting cases yield thermal, field, or photoemission equations. The quantity bF(Eo – E) is the equation of the tangent line to y(E) at E ¼ Em. It is convenient to recast Eq. (275) in terms of a dimensionless integral (Jensen, O’Shea, and Feldman, 2002)   bT
2 ; b ðEo
mÞ; bF Ec ; JðF ; TÞ ¼ C ðEm ÞARLD ðkB bT Þ N ð276Þ bF F where Em is the maximum of the integrand, and ARLD ¼ mqk2B =2p2
h3 ¼ 120.173 amp/k2cm2 is the Richardson constant (Richardson and Young, 1925; augmented by Fowler, 1928 by a factor of 2 to account for electron spin) evaluated using contemporary values of the fundamental constants (e.g., see http://physics.nist.gov/constants). The coefficient C(E) is presumed to be slowly varying (an intuition shown to be reasonable from the analytical models for which an exact evaluation is possible, as well as Airy function approach numerical studies) and of order unity; few have patience for such things given the uncertainty in quantities such as emission area and local work function, so that simply approximating it by unity is an irresistible temptation. Nevertheless, in low field thermionic emission studies, the wave nature of the electron induces ripples captured by C(E(k)) that can be measured (Haas and Thomas, 1968). However, below we succumb to temptation and approximate C(E(k)) by unity. The introduced function N(n,s,x) is represented by dimensionless integral defined by ðu ln½1 þ expðnðz
sÞ N ðn; s; uÞ ¼ n dz; ð277Þ 1 þ expðzÞ
1 where n ¼ bT =bF and s ¼ bF (Eo – m). Eq. (271) is general; it is applied to either field or thermal emission by specifying whether n > 1 or n < 1, respectively. The value of s and Eo in each case is different. A general expression for Eq. (277) will be found in due course, but first, it is pedagogically valuable to investigate two of the three historical current density antecedents in the canonical equations of field, thermionic, and photoemission emission that are based on the image charge potential, namely,

104

KEVIN L. JENSEN

the FN (field) and RLD (thermionic) the barrier maximum pffiffiffiffiffiffiffiffiffiffiequations. Near p ffiffiffiffiffiffiffiffiffiffi V(xo) ¼ m þ F, where xo ¼ Q=F and f ¼ F
4QF , V(x) is well approximated by a quadratic; therefore, y(E) is linear and given by Eq. (265), the slope factor bF for which is smaller than for energies closer to m. What is said about the quadratic barrier bF therefore can be used as a guide to the image charge bF. For example, using the quadratic bF, n is given by !1=2   1=4 2
h2 F3 1 : ð278Þ nquad ¼ p 2m kB T Q In particular, n ¼ 1 for T ¼ 1000 K and F ¼ 1.1 eV/nm. Higher temperatures or lower fields are therefore indicative of the thermal regime, whereas lower temperatures or higher fields are indicative of the field regime. The slope factor bF evaluated from the FN terms is larger, but the qualitative behavior is similar. It is important to note that bF is not beholden to either the FN or quadratic parameterization. It varies depending on where the tangent line to y(E) is taken. Two limits are considered in turn. 1. The Fowler–Nordheim and Richardson–Laue–Dushman Equations Strong fields and low temperatures make n large and signal that tunneling dominates thermal emission. Conversely, high temperatures and weak fields render n small and signal that thermal emission dominates tunneling. The asymptotic limits of Eq. (277) under the assumption that u  s  1 are readily represented as  e
ns ð n ! 0Þ N ðn; s; uÞ ) : ð279Þ n2 e
s ðn ! 1Þ It is then a question of determining the Eo component of s. For n asymptotically small because bF is large, the transmission probability approximates a Heaviside step function in energy with the step occurring at the barrier maximum m þ F; the integrand maximum must therefore occur at (rather, very near to) that energy. Conversely, for n asymptotically large because bT is large, then the supply function vanishes at the Fermi level and the integrand maximum occurs near m and the FN approximation to y is warranted. Thus, asymptotically, 8 > 0 1 ðn ! 0Þ >m þ F < bfn A Eo  ð280Þ mþ@ ðn ! 1Þ > > : Fcfn

ELECTRON EMISSION PHYSICS

105



pffiffiffiffiffiffiffiffiffiffi The ratio bfn = F Fcfn ¼ 2vðyÞ=ð3ð1
yÞtðyÞÞ, where y ¼ 4QF =F, varies from 2/3 to 1 as y varies from 0 to 1. Thus, the value of Eo from the thermal (small n) side is comparable to the field (large n) side. Use of Eq. (280) in Eq. (276) with the approximation for N given by Eq. (279) results in the following asymptotic limits of the general emission equation  JRLD ¼ ARLD T 2 exp½
f=kB T  ðn ! 0Þ


2

JðF ; TÞ ) ð281Þ JFN ¼ ARLD kB cfn exp
bfn =F ðn ! 1Þ These are not the forms encountered in the literature (although they follow naturally from the present analysis). The most commonly given forms are h i pffiffiffiffiffiffiffiffiffiffi qm 2 ð k T Þ exp
F
4QF T =k JRLD ðT Þ ¼ B B 2p2
h3 0 pffiffiffiffiffiffiffiffiffiffiffiffi 1 ð282Þ 3 q 4 2mF 2 @
A JFN ðF Þ ¼ vðyÞ F exp 3
hF 16p2
hFtðyÞ2 The roles of F and T are interchanged in the asymptotic limits, apart from changes in the work‐function–dependent coefficients. A sense of the magnitude of the various terms is useful. For typical thermionic emission conditions of a barium dispenser cathode operating at 1 A/cm2, an extraction grid is held at kilovolt potentials fractions of a millimeter above the emitter surface, which in turn is heated. Using values of F ¼ 1 eV/mm (corresponding to an electric field of 1 MV/m), F ¼ 2 eV, and T ¼ 1300 K implies n  b=cfn ¼ 1=1500. Conversely, field emission from a sharpened Spindt‐type field emitter cone (Spindt et al., 1976) emitting 25 mA from an emission area about (5 nm)2 with F ¼ 7.86 eV/nm, F ¼ 4.41 eV, and T ¼ 300 K, to give n ¼ 13.13. Under such conditions, an array of Spindt‐ type emitters with a packing density of 108 #/cm2 likewise produces a current of 1 A/cm2 (these are ad hoc numbers; arrays driven harder need not be so tightly packed—the hardest‐driven Spindt‐type emitters have achieved more than 1 mA per tip (Schwoebel, Spindt, and Holland, 2003)). The peculiar middle ground near n ¼ 1 can occur when dispenser cathode temperatures get cold or field emitters get (very) hot, circumstances that are not generally encountered (an exception being Schottky emission cathodes; Fransen, Overwijk, and Kruit, 1999), or heated (via high‐intensity lasers) metallic needles subject to high fields (Garcia and Brau, 2001, 2002; Jensen, et al., 2006b). Aside from its historical significance (which is great) and its usage (which is widespread), further discussion of Eq. (282) from a pedagogical view provides diminishing returns; such treatments are replete in the literature and cleaved to with steadfast (and on occasionally unthinking) tenacity, but they are in fact incomplete by virtue of ignoring tunneling in the RLD equation

106

KEVIN L. JENSEN

and thermal emission in the FN equation, although the analyses of Murphy and Good (1956) and Gadzuk and Plummer (1971) are notable counterexamples in efforts to consider thermal‐field emission and provide a thermal correction to the most often used form of the FN equation. Nevertheless, treatments of the middle ground characterized by n  1 are rare. Technological advances rarely leave stones unturned for long, and therefore, the tunneling modifications to thermal emission or the thermal modifications to field emission have a utility apart from the symmetrical beauty of a more general analysis that we shall now develop.

2. The Emission Equation Integrals and Their Approximation In an age of breathtaking desktop computational power, the numerical evaluation Eq. (277) so effortlessly reproduces the FN and RLD equations in the appropriate limits (Hare, Hill, and Budd, 1993; Xu, Chen, and Deng, 2000) that the pursuit of analytic formulas to augment Eq. (281) seems either academic, anachronistic, or obsessive. That is mistaken; numerical methods do not reveal the underlying connection between the two equations hinted at by the formal dependence on F and T evinced in Eq. (281). Early in the twentieth century, the striking similarity of the RLD and FN equations suggested to Millikan and Lauritsen (1928, 1929) that a general form of the current density is J ¼ AðT þ cF Þ2 expð
B=ðT þ cFÞÞ. The actual relation, as shall be seen, bears a subtle beauty well beyond Millikan’s erroneous conjecture. The function N(n,s,u) can be separated into four integration regions that admit of series expansions such that each term in the expansion can be analytically integrated. Therefore Nðn; s; uÞ ¼

4 X

Ni ðn; s; uÞ:

ð283Þ

i¼1

The integrals corresponding to N1 and N2 are field emission dominant and are 8u 9 <ð nðz
sÞ = dz N1 ðn; s; uÞ ¼ n : ez þ 1 ; 8 su 9 ð284Þ <ð ln½1 þ e
nðz
sÞ  = N2 ðn; s; uÞ ¼ n dz : ; ez þ 1 s

107

ELECTRON EMISSION PHYSICS

whereas N3 and N4 are thermal emission dominant and are 8 0 9 < ð ln½1 þ enðz
sÞ  = N3 ðn; s; uÞ ¼ n dz : ; ez þ 1
1 9 8s <ð ln½1 þ enðz
sÞ  = N4 ðn; s; uÞ ¼ n dz ; : ez þ 1

ð285Þ

0

The evaluation of N1 is trivial and gives N1 ðn; s; uÞ ¼ nfUð
sÞ
Uð
uÞg
nðu
sÞlnð1 þ e
u Þ;

ð286Þ

where the Fowler–Dubridge function (Bechtel, Lee, and Bloembergen, 1977; DuBridge, 1933; Fowler, 1931; Girardeau‐Montaut and Girardeau‐ Montaut, 1995) has been introduced and is defined by

UðxÞ ¼

p2 þ 12

ðx 0

lnð1 þ ez Þdz ð287Þ

1 p2 Uð
xÞ ¼ x2 þ
UðxÞ 2 6

although a convenient analytical approximation of reasonable accuracy is given by Jensen et al. (2003b) as 8 x > < e ½1
cb expðca xÞ 2 2 UðxÞ  x þ p
e
x ½1
cb expð
ca xÞ > :2 6

x0 x>0

ð288Þ

where cb ¼ 1
ðp2 =12Þ ¼ 0.17753 and ca ¼ ð1
cb
ln2Þ=cb ¼ 0.72843. Considering typical values across the range of thermal, field, and photoemission processes, the size of u is such that all terms containing e
u are generally negligible. The rationale for partitioning the sums in this manner is that convergent expansions for ln½1 þ z and ½1 þ z
1 for z  1 can be used in the Ni such that the remaining integrals become summations. In each of the integrals, the following replacements are made

108

KEVIN L. JENSEN

ð1 þ ez Þ
1 ¼

1 X ð
1Þkþ1 ekz k¼1

lnð1 þ

enðz
sÞ Þ

¼

1 X ð
1Þjþ1 j¼1

j

e

jnðz
sÞ

ð289Þ

Term‐by‐term integration over the elements of the summations is now possible. The integral for N1 is straightforward and gives N1 ðn; s; uÞ ¼ n2 fUð
sÞ
Uð
uÞg þ n2 ðs
uÞlnð1 þ e
u Þ ) n2 Uð
sÞ

ð290Þ

where the second line results from neglecting u‐dependent terms. The series expansions of Eq. (289) and term‐by‐term integration results in 1 X 1 X  ð
1Þkþj e
ks  1
eðs
uÞðnjþkÞ j ðnj þ kÞ k¼1 j¼1 0 1 1 X kþ1
ks @kA ) ð
1Þ e Z n k¼1

N2 ðn; s; uÞ ¼

ð291Þ

where the k and j terms reflect the series of Eq. (289) with the same index; the second line is obtained by neglecting the u‐dependent terms. For large s, only the k ¼ 1 term survives. The leading order terms of interest for N1 and N2 are independent of u: such will also be the case with N3 and N4, and in all cases it is the same; when u appears, it appears in an exponent with a negative coefficient, and its size indicates that such terms are negligible, and so exponential terms containing (‐u) are summarily neglected. The Z function has been introduced and is defined by

Z ðxÞ ¼

1 X ð
1Þjþ1 j¼1

j ðj þ xÞ

:

ð292Þ

Special cases are Zð0Þ ¼ zð2Þ=2 ¼ p2 =12, where z(x) is the Riemann zeta function, and Z(1) ¼ 2ln(2) – 1. Asymptotic expansions for large and small x are, for large x

109

ELECTRON EMISSION PHYSICS

8 9 = 1 X 1< 1 lnð2Þ
ZðxÞ ¼ x: ð2j þ xÞð2j þ x
1Þ; j ¼1 8 0 9 1 < = 1 x þ 1A 1 ln@4 þ ) 2x : xþ2 ð x þ 2Þ ð x þ 1Þ ;

ð293Þ

where the second line follows from the integral approximation to the series summation, and for small x, an expansion of 1/(jþx) gives Z ðxÞ ¼

1

1X 1
2
j ð
1Þj zð1 þ j Þxj : x j ¼1

ð294Þ

The asymptotic limits are therefore xþ3 ð2zð2Þ
3Þ2 þ x!0 2ðx þ 1Þðx þ 2Þ 2f½6zð3Þ
7x þ 2½2zð2Þ
3g 8 0 9 1 < = 1 x þ 1A 1 ln@4 þ lim Z ðxÞ ¼ x!1 2x : xþ2 ðx þ 2Þðx þ 1Þ; lim Z ðxÞ ¼

ð295Þ

Figure 38 shows the exact value of Z(x) compared to its asymptotically large and small (Eqs. (293) and (294), respectively) approximations. The awkward asymptotic expressions are purposefully constructed to reveal the singular

Z(x)

1

0.1 Z(x) Large x Small x 0.01 0

0.8

1.6 2.4 ln (1 + x)

3.2

4

FIGURE 38. Comparison of Z(x) to its asymptotic limit formulae [Eqs. (293) and (294)].

110

KEVIN L. JENSEN

behavior at x ¼
1. Continuing, the integral for N3 can be recast using ðez þ 1Þ
1 ¼ 1
ðe
z þ 1Þ
1 and ½j ðk þ jnÞ
1 ¼ ðjkÞ
1
n½kðk þ jnÞ
1 to obtain 1 X

ð
1Þkþ1 e
kns ZðknÞ:

ð296Þ

8 9 0 1 1 <
ks = X kA kþ1 e
kns @ þ ne Z
þ n ð
1Þ Z ð
knÞ : ; n n k¼1

ð297Þ

N3 ðn; s; uÞ ¼ Uð
nsÞ
lnð2Þnlnð1 þ e
ns Þ þ n2

k
1

The final integral is N4 ðn; s; uÞ ¼ lnð2Þlnð1 þ

e
ns Þ

Different terms survive depending on whether ns or s is larger, thereby leading to the RLD and FN equations. 3. The Revised FN and RLD Equations Consider now the RLD and FN‐like limits, which correspond to s  ns  1 and ns  s  1, respectively. In the RLD limit (small n), N1 and N2 are negligible, only the k ¼ 1 terms survive in the series expansions, and the U and log functions can be replaced by their leading order terms (e.g., ln(1þx)  x). In the FN limit (large n), N3 is negligible, N1 is replaced by its leading order terms, and the k ¼ 1 terms survive in the series expansion of N3 and N4. When going through the mechanics of finding the dominant terms, it becomes apparent, after a bit of regrouping, that to leading order N ðn; s; uÞ ! N ðn; sÞ and 0 1 1 N ðn; sÞ ¼ S@ Ae
s þ SðnÞe
ns ð298Þ n SðxÞ  1 þ x2 fZ ðxÞ þ Z ð
xÞg In other words, N naturally separates into two parts: the part containing e
ns is the thermal‐like term, and the part containing e
s is the field‐like term, so called because their asymptotic limits give rise to the canonical RLD and FN equations, respectively. Explicitly, the revised FN‐RLD equation can be written 

ðJF =n2 Þ þ JT ðn < 1Þ JF þ n2 JT ð n > 1Þ ðnÞe1
ns JT  ARLD ðkB bT Þ
2 S0

J ðF ; T Þ ¼

1 JF  ARLD ðkB bF Þ
2 S@ Ae
s n

ð299Þ

ELECTRON EMISSION PHYSICS

111

The symmetry between field emission and thermionic emission is made a bit more manifest by using the series expansion form of S(x) to show 8 9 < = 7 31 NRLD ðn; s; uÞ ¼ 1 þ zð2Þn2 þ zð4Þn4 þ zð6Þn6 þ . . . e
ns : ; 4 16 8 9 ð300Þ < = 7 31
4
6
2 2
s NFN ðn; s; uÞ ¼ 1 þ zð2Þn þ zð4Þn þ zð6Þn þ . . . n e : ; 4 16 of which the leading terms have been anticipated by Eq. (279). Using the explicit forms for the Riemann zeta functions and utilizing Eq. (276), it follows that the revised FN and RLD equations become (remember the temptation to approximate the coefficient C by unity) 0 JRLD ðF; TÞ ¼ ARLD ðkB bT Þ


2 @

0 12 0 14 1 p2 @bT A 7p4 @bT A 1þ þ þ . . .Aexpf
bT ðEo
mÞg 6 bF 360 bF

0 12 0 14 1 2 4 p b 7p b @ F A þ . . .Aexpf
bF ðEo
mÞg JFN ðF ; TÞ ¼ ARLD ðkB bF Þ
2 @1 þ @ F A þ 6 bT 360 bT 0

ð301Þ As a historical note, the first correction term in parentheses for JFN is, using the FN representation for bF, the same as an expansion of the thermal correction term found by Murphy and Good (1956) in their Eq. (77). The term Eo changes from thermal to field emission conditions. It is known from the thermal and field regimes that Em is at the barrier maximum or the chemical potential, respectively, that is Em ðn 1Þ ¼ m þ f Em ðn  1Þ ¼ m

ð302Þ

from which Eo is given by (as seen in the FN and RLD equations) Eo ðn  1Þ ¼ m þ

2vðyÞ F 3tðyÞ

ð303Þ

Eo ðn 1Þ ¼ m þ f Restricting n to very much larger or smaller than 1, as done here, is slight overkill, as they generally work reasonably well under less stringent demands—but it is precisely the region where n is near 1 that difficulties arise; these are explored next.

112

KEVIN L. JENSEN

The symmetry of Eq. (301) is appealing, although it should be observed that (1) the term Eo differs between the FN and RLD limits, and (2) while the two expressions converge for n ¼ 1, neither is correct at that point. The point where n ¼ 1 constitutes the transition region, in which the current integrand peak shifts from near the barrier maximum to near the chemical potential as n advances from below unity to above it. Near the n ¼ 1 transition region, all the integrals entailed by N1 through N4 contribute, and N4 in particular contains a term whose denominator goes as (n – 1)
1. To leading order, Ntrans ðn; sÞ ¼

nðe
s
e
ns Þ þ e
s þ ðn
1Þ2lnð2Þe
s : n
1

ð304Þ

The vanishing denominator is therefore offset by a vanishing numerator, so that L0 Hoˆpital’s rule may be used. Therefore, when n ¼ 1, the ‘‘transition’’ current density is Jtrans ðF ; TÞ ¼ ARLD ðkB bT Þ
2 ½bF ðEo
mÞ þ 1expf
bF ðEo
mÞg;

ð305Þ

which is larger than Eq. (301) when bF ¼ bT for a given Eo. Therein lies a difficulty in the implementation of Eqs. (301) and (305). From Equations (302) and (303), the value of Eo changes depending on the asymptotic limit of n. To use Eqs. (301) and (305), two questions must be addressed. First, how shall Eo be calculated when n is of order unity? And second, how is bF to be determined under general conditions? It is numerically evident that the optimal tangent line to y(E) should be taken at the maximum of the current integrand, Em; errors in the integrand away from this energy are exponentially damped by either the transmission probability or the supply function. Thus, n ¼
bT f]E yðE ¼ Em Þg
1 bT Eo ¼ bT Em þ nyðEm Þ

ð306Þ

For thermionic emission conditions, it is clear that Em lies close to m þ F. For energies above the barrier maximum, the linear extension approximation to y(E) [see the discussion following Eq. (249)] ensures that n in Eq. (306) is trivially evaluated using Eq. (275), and it therefore follows that !1=2   1=4 2
h2 F3 1 ntherm ¼ ; ð307Þ p 2m kB T Q for example, n ¼ 0.01 for F ¼ 10
3 eV/nm and T ¼1047 K. Observe that n scales as n / F 3=4 for n < 1. It is natural to inquire if n for n > 1 follows a similar power law behavior—as shown by direct numerical evaluation (Jensen and Cahay, 2006) and as shall be demonstrated in the following

ELECTRON EMISSION PHYSICS

113

text, it does. However, the optimal evaluation of the power for nfield benefits most from good approximations to the elliptical integral functions v(y) and t(y) and therefore must await the introduction of the Forbes approximation to v(y) below. In the meantime, numerical means suffice to consider the performance of the revised FN‐RLD equation. It is a straightforward matter [using the form of T(E) provided by Eq. (251) with C(k)  1 and y evaluated using the WKB AUC method of Eq. (256)] to find the location of the current density integrand maximum, Em, by bracketing and bisection. Having found Em numerically, n is evaluated using Eqs. (306) and (256). The behavior of n(F) for copper parameters at 800 K is shown in Figures 39 and 40 along with the thermal and field power‐law relations ntherm ¼ 1:164F 3=4 nfield ¼ 0:661F 0:948

ð308Þ

where the coefficients and pFN ¼ 0.948 are determined from the F ¼ 0.02 eV/ nm (thermal) and F ¼ 10 eV/nm (field) data points. Clearly, therefore, the current integrand maximum migrates from near the barrier maximum (m þ F) to near the Fermi level (m) as the field increases (Dolan and Dyke, 1954; Gadzuk and Plummer, 1971; Jensen, O’Shea, and Feldman, 2002; Murphy and Good, 1956). The next question is: How does the shape of the integrand change during the same evolution? In Figure 41 the location of the integrand maximum is bracketed by the two values where the integrand is 1% of its maximum (designated Emax and Emin for the larger and smaller energy, respectively) for the same conditions as in Figures 39 and 40. Several features are noticeable. First, in the thermal regime, Em remains fairly close to the barrier maximum (m þ F). Second, in the field regime, Em is close to, but generally not at the Fermi level, and at high fields, Em can be below the Fermi level: when the tunneling electrons are replaced by electrons from higher energies, the excess energy appears as heat in a process called Nottingham heating (Ancona, 1995). Conversely, at low fields, tunneling electrons primarily come from above the Fermi level and cooling occurs. Third, as shown in Figure 42, the energy full width at half maximum (EFWHM) increases substantially near the n ¼ 1 region; the locations where n differs from 1 by less than 2% are shown by the open circles. For almost all fields of technological interest (F  10 eV/nm), EFWHM is largest in the n  1 transition region. For copperlike parameters, compared to standard field or thermionic conditions of 4 eV/nm and 300 K (field) or 0.05 eV/ nm (i.e., 5 MV/m) and 1500 K (thermionic), the integrand for the n  1 transition region (1.36 eV/nm and 800 K) is substantially broader than either the field or thermionic cases (the total current density in each case is

114

KEVIN L. JENSEN

10 n ntherm

0.661 F0.948

n = bT/bF

nfield n=1

Field regime

1 Thermal regime Φ = 4.6 eV m = 7.0 eV T = 800 K F = 1 eV/nm

1.164 F3/4 0.1

0.1

1 Field [eV/nm]

10

FIGURE 39. Behavior of the slope factor ratio (n) as a function of field for copper-like parameters and moderate temperatures. The n = 1 line demarcates the transition region between thermal (n < 1) and field (n > 1) conditions.

1.6 nexact

0.661 F0.948

n = b T/b F

ntherm nfield

1.2

n=1

0.8

1.164 F3/4 0.4 0.5

1

1.5

2

2.5

Field [eV/nm] FIGURE 40. Close-up of the n = 1 region of Figure 39.

substantially different; the parameters are chosen for pedagogical rather than pragmatic reasons). The transition is revealed more readily by holding the field fixed at 1.36 eV/nm and raising the temperature as shown in Figure 43a (the temperature variation of the chemical potential is ignored) for the temperatures 300 K, 700 K, 800 K, 900 K, and 1500 K for copper parameters; the 900 K case shows a broad distribution in particular. Figure 43b shows the behavior of the (normalized) integrand as the temperature is adjusted

115

ELECTRON EMISSION PHYSICS

12

Energy [eV]

10

Thermal regime

Field regime

8 6 4

Emax

µ+φ

Em

µ

Emin

n=1 Regime

T = 800 K 2 0.1

1

10

Field [eV/nm] FIGURE 41. Behavior of the full-width-at-half-max (FWHM) separation as a function of field for the same parameters as Figure 39.

12 |n(F)-1| ≤ 0.02

1.6

10

1.2 8 0.8

EFWHM [eV]

Energy [eV]

2.0

0.4

6 0.1

1

10

0.0

Field [eV/nm] FIGURE 42. Location of the integrand maximum Em and the full-width-half-maximum energy width as a function of field for the parameters of Figure 39.

upward for constant field. The nature of the broadening of the integrand as the peak shifts through the transition region bounded by m < Em < m þ F is readily apparent. It remains to compare the performance of the revised FN‐RLD equation where n is found numerically (an analytical method will be presented after the Forbes approximation to v(y) is introduced). First, the comparisons are made for regimes in which the FN and RLD equations are known to perform well. Figure 44a compares the revised FN‐RLD and the standard FN and

116

KEVIN L. JENSEN

(a)

1.2

dJ(E)/dJmax

1

T = 300 K F = 4 eV/nm

T = 1500 K F = 0.05 eV/nm T = 800 K F = 1.36 eV/nm

0.8 0.6 0.4

Cu

0.2 0

7

6

8

9

10

11

12

Energy [eV] (b) 1.2

dJ (T)/dJmax

1

700 K 300 K 800 K

900 K 1500 K

0.8 0.6 0.4 Cu

0.2 0 7

8 9 Energy [eV]

10

FIGURE 43. (a) Behavior of the current density integrand for thermal (right), field (left) and mixed (middle) conditions. (b) Same as (a) but showing the intermediate cases. Note the width of the curve labeled “900 K.”

RLD equations. In the standard FN equation, the Spindt quadratic approximation v(y) mentioned after Eq. (262 ) and t(y) ¼ 1.0566 is used (the reason for the designation ‘‘standard’’ is because of the wide use of this form in inferring work function from slope on an FN plot). Not surprisingly, the agreement is excellent. Second, for the fictitious case where a low work function coating is present with work function of F ¼ 1.8 eV (but otherwise the copper parameters are retained), as shown in Figure 44b, the standard RLD equation is generally adequate to within 15%, whereas the revised FN‐RLD is good to better than 1%. The interest comes, however, for moderate temperature and field regimes, as in Figure 44c, where the temperature is high (but not as high as for thermionic cathodes) and the work

117

ELECTRON EMISSION PHYSICS

(a) 1011 Cu under field conditions (T = 300 K, Φ = 4.6 eV)

Current density [A/cm2]

109 107 105 103 101 10−1

Numerical Revised FN-RLD Standard FN (Spindt aprox)

10−3 10−5 10−7 10−9

1

10 Field [eV/nm]

Current density [A/cm2]

(b) 107

Cu w/coating under thermal conditions (T = 1400 K, Φ = 1.8 eV)

106

Numerical Revised FN-RLD Standard RLD

105 104 103 102

(c) Current density [A/cm2]

1011

0.1 Field [eV/nm]

1

Cu under mixed conditions (T = 800 K, Φ = 1.8 eV)

109 107 105 103 101 10−1 0.1

Numerical Revised FN-RLD Standard FN Standard RLD

1 Field [eV/nm]

10

FIGURE 44. Performance of the revised Fowler-Nordheim–Richardson-Laue-Dushmann equation [Eq. (301)] (a) compared to the most commonly used forms of the Fowler–Nordheim equation for copperlike parameters; (b) compared to the most commonly used forms of the Richardson-Laue-Dushmann equation for cesium on copperlike parameters; (c) compared to mixed conditions challenging the Fowler–Nordheim and Richardson-Laue-Dushmann equations for cesium on copperlike parameters. Note the high-field behavior.

118

KEVIN L. JENSEN

function is low (by comparison to field emitters). Here the superiority of the revised FN‐RLD equation is manifest where it is expected to be better, but also in the high field region where the superiority of the FN‐RLD equation is likewise evident. Before considering a general thermal‐field equation it is profitable to determine the performance of the FN and RLD equations (through the eyes of the FN‐RLD equation) in practice. F. The Revised FN‐RLD Equation and the Inference of Work Function From Experimental Data 1. Field Emission The common motivation for representing v(y) as a linear function in y2 and t(y) as a constant is that the FN coordinate lnðJ=F 2 Þ is linear in 1/F, or

B ln J=F 2  A
: F

ð309Þ

Using the approximations vðyÞ ¼ vo
y2 and tðyÞ ¼ to , where vo and to are constants independent of field, then 4vo pffiffiffiffiffiffiffiffiffiffiffiffi3 B 2mF 3h 0 1 sffiffiffiffiffiffiffi ð310Þ 16 2m q A þ ln@ A Q 3
h F 16p2
ht2o F For example, consider the ‘‘prediction’’ of the work function using as data points the evaluation of current density using the revised FN‐RLD equation and copperlike parameters in the range 2 eV/nm < F < 10 eV/nm. Inferring the work function using Eq. (310) (with vo ¼ 0.93685 and to ¼ 1.0566) from the calculated slope B ¼ 63.472 eV/nm indicates F ¼ 4.6162 eV, very close to the value of 4.60 eV used in the simulation of J. Backing out theoretical parameters from ad hoc simulations is scholasticism, even though it indicates the accuracy of an approach. Of greater interest is to what accuracy the work function can be determined from actual field emission data based on the FN equation. Of the several existing methods to measure work function (Haas and Thomas, 1968), estimating F from the value of B is widely used (a good example is Swanson and Strayer, 1968), and therefore, showing how it fares is useful. In early studies of field emission from tungsten wires, data with and without barium adsorbed onto the apex of the emitter were taken by Barbour et al., (1953; Figure 3 of Barbour et al. is represented in Figure 45; the straight line fits are explained below). The tungsten needle geometry allowed for fields greater than 1 GV/m to be

119

ELECTRON EMISSION PHYSICS

ln { I/V2 [Amp/Volt2]}

−20 Pulsed current Measurements

−24 −28

1

2

3

4

−32 −36

Direct current Measurements

1

2

3

104/V [Volt] FIGURE 45. Fit to the data of figure 3 of Barbour et al., (1953) for a clean tungsten emitter (1) and the same emitter with increasing amounts of surface coverage by barium.

generated at the apex. An immediate complication is that current is measured as a function of potential differences between cathode and anode, whereas the FN equation relates current density to field at the emission site. Why the naive presumption that current is proportional to current density by an area factor (and voltage to field by a ‘‘beta’’ factor) is addled has been subject of much work (Forbes, 1999a; Forbes and Jensen, 2001; Jensen and Zaidman, 1995; Nicolaescu, 1993; Nicolaescu et al., 2001, 2004; Zuber et al., 2002). The problem is twofold: the field varies considerably over the sharpened structures required to obtain significant field enhancement, and the rapid variation of field over the surface means that the emission area changes depending on field strength on‐axis and its variation off‐axis. Barbour et al. (1953) note this in their analysis of the emission data, but assume that the emission area is constant in order to facilitate the analysis. A simple model of the impact of both field enhancement and emission area can be used to obtain a refined analysis compared to that of Barbour et al. (1953). Consider emission from a hemisphere of radius a. It is a simple problem in electrostatics to show that the field along the surface of such a hemisphere on a grounded plane is given by F ðyÞ ¼ 3ðV =DÞcosðyÞ for a sufficiently distant anode held at a potential V a distance D away. The field enhancement factor at the apex of the hemisphere is therefore (3/D), that is, Ftip ¼ ð3=DÞV ¼ bo V (reflecting the nomenclature beta factor: the proliferation of quantities referred to by the b symbol induces the ‘‘o’’ subscript (o) to avoid confusion with the temperature and field slope factors).

120

KEVIN L. JENSEN

It follows that the current from the hemisphere can be written as the product of an ‘‘effective’’ area and apex current density, or emission

I ðV Þ ¼ barea Ftip J Ftip where

2pa2 barea Ftip ¼

J Ftip

ð p=2 0

J ½F ðyÞsinðyÞdy:

ð311Þ

The approximation that t(y) is constant and v(y) ¼ vo – y2 is very convenient, and from that approximation, it follows

barea Ftip ¼ 2pa2

8 9 < B = BFtip exp
ðx
1Þ x2 dx  2pa2

2 : : Ftip ; 0 B þ 2Ftip

ð1

ð312Þ

An area factor for a hemisphere is but the simplest approximation possible, but considering either an ellipsoidal or hyperbolic emitter geometry does not change the overall conclusion that the area factor is linearly dependent on field; using F ¼ 4.5 eV and Eq. (262), then Ftip/B ¼ 0.082 for Ftip ¼ 5 eV/nm. The weak field dependence of the denominator changes in the RHS of Eq. (312) for the geometries characteristic of wires and field emitter arrays, respectively, but it is still found that for field emission from metals in general, B is sufficiently larger than Ftip that to a good approximation, barea scales linearly with apex field. A field‐dependent area factor undercuts the common practice to plot current‐voltage data in FN‐like coordinates of (1/V) versus ln(I/V2) and to infer from linearity of the resulting plot that field emission is occurring and the coefficient of 1/V gives information about the work function. Given experimental uncertainty, ln(I) versus 1/V is likewise fairly linear—and given the variation in emission area with apex field, there is no reason to expect an FN‐like coordinate lnðI=V 2 Þ to occupy any privileged role. In fact, given the behavior of barea, the proper coordinates are lnðI=V 3 Þ versus 1/V (if the statistics of dissimilar emitters is considered, then another power of V appears in the denominator of the log function; Cahay, Jensen, and vonAllmen, 2002; Jensen and Marrese‐Reading, 2003) Nevertheless, in the literature, when the slope factor B is referenced (typically to infer work function), it is under the approximation that lnðI=V 2 Þ is linear in 1/V. An example in the case of carbon nanotubes, which have small emission areas and sharp apexes indeed, is the work of Fransen et al. (1999). It is possible to reconcile the standard approach with the physics, and such is the approach taken here. Let Vo be a particular voltage for which the current is Io and the apex field is Fo ¼ bo Vo , so that IðV Þ ¼ barea ðFo ÞðF =Fo ÞJFN ðF Þ. Then it follows

121

ELECTRON EMISSION PHYSICS

lnfI ðV Þ=V 2 g  A0
B0 ¼

B0 V ð313Þ

B þ Vo bo

  A0 ¼ 1 þ A þ ln b2o barea ðbo Vo Þ Therefore, values of B0 extrapolated from experimental data are related to the work function (assuming it is unique and not a compilation of averaged values over different crystal planes) by ( F¼

9b2o
h2 0 2 ½B
V o  32mv2o

)1=3 :

ð314Þ

If work function changes are occurring [as when barium is being deposited on the tungsten needle, as Barbour et al. (1953) did], then it follows that F1 ¼ F2



B1 0
Vo B2 0
Vo

2=3

:

ð315Þ

Reconsider now the data of Barbour et al. in FN coordinates. The B0 values of the linear fits of lines 1–4 are 145.5, 89.91, 72.23, and 55.02 kV, respectively. Using as the reference point for line 1 values of Io ¼ 0.6457 mA at Vo ¼ 7981 V, an assumed F for tungsten of 4.5 eV, and A0 ¼
14.22, it is inferred that bo ¼ 4441 q/cm and barea(boVo) ¼ 3.930 10
10 cm2, values comparable but not equal to those of Barbour et al. The other lines correspond to progressively greater amounts of barium deposited on the surface. A partial coverage of alkali and alkali earth metals on other metals tends to lower the work function, so that the lines 2–4 reflect decreasing values of the effective work function as the surface coverage of barium increases. We infer from the linear fits and the work function of clean tungsten that the value of F for lines 2–4 are 3.19 eV, 2.71 eV, and 2.20 eV, respectively, which are comparable to (but smaller than) the values obtained by Barbour et al. because of the present approximation of a field‐dependent area factor. 2. Thermionic Emission For thermionic emission, the variation of current density with temperature allows for the estimation of work function, as the Richardson coordinate lnðJ=T 2 Þ is linear in 1/T. Here, complications such as area factors do not arise. Where F ! T in Eq. (309), we have

122

KEVIN L. JENSEN

pffiffiffiffiffiffiffiffiffiffi 1 F
4QF kB 0 1 2 qmk BA A  ln@ 2p2
h3 B

ð316Þ

Compare the work function evaluated from the slope of current density on an RLD plot using Eq. (316) when the current is given by the revised FN‐RLD equation for parameters somewhat at the edge of generic parameters (e.g., an applied field of 20 MV/m and a work function of 1.8 eV, as suggested by Figure 46). From a slope of 1.641/kB, the work function is inferred by Eq. (316) to be 1.811 eV, quite close to the input value. Similarly, the numerical intercept is close to the theoretical value of ln(120.17 A/cm2K2) ¼ 4.790. Inferring work function from experimental data, however, is the challenge. One difficulty is that the work function is temperature dependent (i.e., F(T) ¼ Fo þ aT) in addition to its dependence on crystal face. Since many experimental metal emitters are polycrystalline and require high temperatures to achieve significant current, the inference of a single or well‐defined work function from experimental data is problematic. Current density is inferred from current and a presumed emission area, but even correcting for area results in A values that differ from theoretical predictions. However, when such factors are corrected for by carefully designed experiments, an estimate of F from the slope of a Richardson plot can indicate

ln { J / T 2 }

−8

y(x) = 4.734 – 1.641 x

−12

−16

−20

Field = 20 MV/m Work function = 1.6 eV 81

01

21

14

1/kB T FIGURE 46. A hypothetical data set created using the Revised Fowler-Nordheim–RichardsonLaue-Dushmann equation so as to compare the accuracy of inferring work function from a Richardson plot.

123

ELECTRON EMISSION PHYSICS

yA = 4.69 − 4.55/ kBT

ln {J/ T2 [A/cm2K2] }

−24

yB = 6.52 − 4.70/ kBT

−27

−30 6.4

6.8

7.2

7.6

1/(kBT [eV]) FIGURE 47. Data considered by Shelton (1957) in the determination of work function from Richardson plots.

work function with some accuracy. An effort to extract work function estimates, allowing for such complications (and others), was undertaken by Shelton (1957). Shelton’s data (shown in Figure 47) give from the slope a naive estimate for the work function for tantalum to be 4.55 eV, but correcting for the work function variation with temperature reduces the value to 4.35 eV, close to the accepted polycrystalline value of 4.25 eV (Weast, 1988). 3. Mixed Thermal‐Field Conditions Under mixed thermal‐field conditions, estimating work function from slopes on RLD and FN plots becomes problematic. At low fields, thermal emission compromises the slope on an FN plot in a manner suggested by Figure 48 so that, apart from the disagreement introduced by a low work function, changes are introduced by an increasingly nontrivial thermal component as the field declines. In such cases, a comparison to numerical evaluations of current density is preferable. It is then a question of what complications can arise, and as expected, complications do arise with the procedure of comparison. Gadzuk and Plummer (1973) describe how total energy distribution (TED) measurements are affected by the finite energy resolution of energy analyzers, and therefore, the energy distribution appears to be smeared out. For example, even a zero‐temperature energy distribution, which would in principle have a sharp edge, nevertheless has a broadening of the distribution near the Fermi level that appears similar to the effects of a raised temperature. We therefore cannot expect a priori an exact

124

KEVIN L. JENSEN

30 25

ln{J/F2}

20 15 10 Numerical FNRLD FN

5 0 1

10

100

1/(F [eV/Å]) FIGURE 48. Departures from the Fowler–Nordheim relation for simulated data based upon the Fowler-Nordheim–Richardson-Laue-Dushmann equation.

correspondence between theoretical models predicated on simple emission calculations with difficult‐to‐obtain energy distribution measurements (even though the total current measurements can be good, representing as they do an integration over the energy distribution). An overall agreement in the qualitative features is satisfactory. A word of caution is necessary; we have not made a distinction between the TED and the normal energy distribution; the latter represents the distribution of our designation Ez. The difference is both nontrivial and important (Young, 1959; Young and Mu¨ller, 1959). A measured field emission distribution measures the TED corresponding to E ¼
h2 k2 =2m, whereas the normal energy distribution is the outcome of looking at the passage of the longitudinal momentum component through the 1D barrier and is therefore Ez ¼
h2 k2z =2m. The distinction has been hidden heretofore because of the focus on 1D emission equations—the TED benefits from a moments‐based analysis (which is not treated here but is discussed later). The discussion here blurs the distinction between Ez and E, although comments about the normal energy distribution will have analogs for the TED (and so the z subscript will not be used on E); a correct analysis is well summarized by Gro¨ning et al. (1999, 2000). Let the energy analyzer have a distribution of energies S(E
Eo) that it accepts when measuring the particle count at an energy Eo; for example, S may be Gaussian of the form S ðE Þ ¼ ð2pgÞ
1=2 expð
E 2 =2gÞ. Thus, the particle count per unit area per unit time (proportional to the current density integrand) for the energy Eo is

ELECTRON EMISSION PHYSICS

d 1 NðEo Þ ¼ dt 2p
h

ð1
1

S ðE
Eo ÞT ðE Þfo ðE ÞdE:

125

ð317Þ

By the normalization of Eo, the integration of dN/dt over Eo reproduces the current density (to within a factor of the electron charge). The more sharply S is peaked (the sharper the energy resolution), the more the experimental results resemble the theoretical energy distribution of emitted electrons. Consider two examples: first, the case of simple Richardson‐like (thermionic) emission, and second, a more Gaussian‐like distribution characteristic of the situation n ¼ 1. For the thermionic case, TðEÞfo ðEÞ / yðE
m
FÞexpð
bT ðE
mÞÞ, and so the integration over the Gaussian form of S in Eq. (317) is readily performed and is 9 8 ð1 < = d 1 ðE
Eo Þ2 NRLD ðEo Þ ¼
bT ðE
mÞ dE exp
p ffiffi ffi 3=2 ; : dt 2g ð2pÞ
h g mþf

8 3 2 39 2 = 1 < gb þ m þ f
E 1 o 5 exp4 gb2
bT ðEo
mÞ5 1
Erf 4 T pffiffiffiffiffi ¼ ; 4p
h: 2 T 2g

ð318Þ where Erf(x) is the error function. The presence of the error function complicates matters, but the effect is a progressive blunting of the sharpness of the emitted electron distribution to a more Gaussian‐like shape; as g becomes larger, the large argument approximation to the error function can be invoked, and it can be shown that " # rffiffiffiffiffiffi d g ðEo
m
FÞ2
1 NRLD ðEo Þ  ðb g þ m þ F
Eo Þ exp
bT F þ ; dt 2p T 2g ð319Þ which demonstrates that as the resolution of the detector becomes progressively worse (g1/2 becomes progressively larger), the measured distribution becomes more Gaussian rather than the characteristic MB‐like behavior. Figure 49 shows the effect on a theoretical distribution for dispenser cathode conditions for various values of g. In Figure 50, the impact of Eq. (317) with g ¼ 0.1 eV2 on a (normal) energy distribution suggested by the experimental conditions of Gadzuk and Plummer (1973) are shown. Note that one of the effects is to make the ‘‘thermal tail’’ appear to be at a higher temperature than is the case. Therefore, when comparing the energy distributions with experimental data, the impact of the resolution of the energy analyzer must be considered in the analysis. Here, a measure of the success of the theory is whether the qualitative dependence is captured—as it is.

126

KEVIN L. JENSEN

Current integrand [a.u.]

1.5 m = 8.0 eV Φ = 1.8 eV F = 10 MV/m T = 1300 K

1.0

γ1/2 0.00 0.05 0.10 0.20

0.5

0.0 9.2

9.4

9.6 9.8 10 Energy [eV]

10.2

10.4

FIGURE 49. Smoothing out of the normal emission distribution with a Gaussian energy analyzer [Eq. (317)].

Current integrand [a.u.]

10−9 10−10

F = 3.50 eV/nm

Φ = 4.8 eV m = 8.0 eV T = 1570 K γ = 0.1 eV2

10−11 10−12 10−13 F = 2.19 eV/nm

10−14 10−15

5

6

7

8 9 10 Energy [eV]

11

12

13

FIGURE 50. Effect of Gaussian energy analyzer on emitted distribution (lines constitute no energy analyzer, symbols are effects of an energy analyzer—both are generated from theoretical models).

Consider, then, conditions such that n transitions through unity, and compare the theoretical current density integrand to the measurements of Gadzuk and Plummer (1973). The criteria for a good comparison are that fields taken in the same proportion as the voltages considered experimentally give rise to similar qualitative relations for the energy distributions. As shown in Figure 51, the theoretical model bears a relation to the experimental trends and largely accounts for the evolution from field to thermal conditions.

127

ELECTRON EMISSION PHYSICS

7 V [V]/F [eV/nm] 1600/3.50 1200/2.63 1000/2.19 800/1.75 600/1.31 500/1.09

6 5

6 5 4

4

3

3

2

2

1

Current integrand (a.u)

Measured charge (a.u.)

7

0 1

−2

−1

0

1 2 E-EF [eV]

3

4

5

FIGURE 51. A comparison of the theoretical energy distributions (lines demarcated by field) with the experimental distributions from Gadzuk and Plummer (1971, 1973) (symbols demarcated by voltage). Relative ordering of numbers reflect relative position of lines.

4. Slope‐Intercept Methods Applied to Field Emission In the previous section on inferring work function from the FN relation, it was explicitly assumed that the field was related to the applied voltage by a scale factor (the ‘‘beta’’ factor), but it was also implicitly assumed that the current is related to the current density by an emission area. A general argument was presented to show that the emission area must be field dependent—but the usage of the slope and intercept parameters is more useful than simply that; it can illuminate the nature of changes that occur on the emitter during conditioning, especially if a model of the field enhancement and area factors can be developed as done by Mackie et al. (2003) and Charbonnier et al. (2005). The present treatment is similar in intent but differs in detail. Complications associated with simultaneously saying something intelligent about work function and field enhancement have been capably treated by Gro¨ning et al. (1997, 1999, 2000). Let us reconsider the field‐dependent area factor analysis behind Eq. (313) and use it to express current versus voltage using current density versus applied field relations. For Spindt‐type field emitters, where the voltage in question comes from a close‐proximity gate, let the relationship between apex field F and gate voltage V be given by F ¼ bg V , where the g subscript (g) identifies that it is the gate that is primarily responsible for the apex field rather than a distant anode. Next, explicitly separate out the field dependence of the area factor. Combining the large B limit of Eq. (312), using Eq. (310) it follows that

128

KEVIN L. JENSEN

9 8 ffiffiffiffiffiffiffi < 16 s2m

3 3qa2 4vo pffiffiffiffiffiffiffiffiffiffiffi3ffi= pffiffiffiffiffiffiffiffiffiffiffiffi
IFN ðV Þ  bg V exp
Q 2mF : ; : 3
h F 3
hbg V 64p 2mF5 vo t2o ð320Þ It is clear, therefore, that Eq. (320) can be transformed into a linear relationship where the slope (s) and intercept (zo) factors are defined according to the relation   IFN ðV Þ s ln ð321Þ ¼
þ zo V3 V and are determined from experimental data. Thus, a comparison between Eqs. (320) and (321) uniquely determines the apex radius and the work function if the slope and intercept are known from the relations pffiffiffiffiffiffiffiffiffiffiffiffi 4vo 2mF3 s 3
hb g 2 3 sffiffiffiffiffiffiffi ð322Þ 3qa2 b3g 16Q 2m 4 5 pffiffiffiffiffiffiffiffiffiffiffiffi þ ln zo  3
h F 64pvo t2o 2mF5 In contrast to claims (implicit or otherwise) occasionally made in the literature, Eq. (322) does not show that the slope on an FN plot gives work function but rather that the slope is proportional to F3=2 =bg and the enhancement factor must be considered, the methods of Gro¨ning et al. being a case study in point (Gro¨ning et al., 1999, 2000). In practice, the application of Eq. (322) is fraught with complications. At the apex of a field emitter, more than one crystal face can be exposed, and crystal faces have different work functions; contamination and oxides can impart features of their own or even reduce the effective emission area; emission can come off‐axis, whereas the theoretical model above presumes on‐axis emission from a rotationally symmetric surface; migration of material can occur; and so on. An extensive study of several of these effects was done by Dyke et al. (1953) in greater detail than allowed by the present treatment. Nevertheless, it is of pedagogical value to see if expectations borne of Eq. (322) are manifest in experimental data. The variation of work function F can be addressed by considering an effective, or averaged, work function over the apex. The geometry factor bg is a bit more difficult as it depends on the particulars of the emitting surface. An approximation, based on the hyperbolic model of a Spindt‐type emitter, suggests that to leading order, the field enhancement factor is inversely proportional to tip radius (simple models, such as needles, also give an

129

ELECTRON EMISSION PHYSICS

inverse relationship to tip radius to leading order). Other factors, such as cone angle and gate radius, contribute additional factors beyond our scope here. The simplest approximation is to use a polynomial (quadratic) fit of abg; based on the hyperbolic model and expanded about a ¼ 10 nm, we use

bg 

ð11:654 þ aÞð59:224
aÞ ; 2429:1a

ð323Þ

where a is in units of nanometers and bg in units of 1/nanometer (parenthetically, note that the gate radius and cone angle are implicitly assumed to be 0.5 mm and 15 , respectively, in the evaluation of the numerical parameters in Eq. (323)). Of course, there are higher‐order effects, the neglect of which will affect, for example, the value of the work function converged on, but these considerations are ancillary to the present treatment. Now consider the data from Figure 2 of Schwoebel, Spindt, and Holland (2003) showing the changes wrought on single‐tip Spindt‐type field emitters subject to conditioning (reproduced in Figure 52 and recast in Figure 53) in FN coordinates from which the slope and intercept factors are ascertained. The ‘‘conditioning’’ entailed controllably heating the field emitter tips by drawing intense currents; the heat was sufficient to smooth and recrystallize the apex by surface diffusion, as well as to drive off contaminants by thermal desorption. Surface self‐diffusion tends to come into equilibrium with applied field for a particular apex configuration (Barbour et al., 1960;

Current [Amp]

10−5

10−6

I1 FN I2 FN F1 FN F2 FN

10−7 60

80

100

120

140

160

I1 I2 F1 F2

180

Voltage [V] FIGURE 52. Preconditioning and post-conditioning current-voltage plots of the emitter tips examined by Schwoebel et al. (2003). Symbols are experimental data; lines are based on the theoretical models examined in the text.

130

KEVIN L. JENSEN

ln{ I/V3 [A/V3]}

−26 −28 FNI1 Fit FNI1 FNI2 Fit FNI2 FNF1 Fit FNF1 FNF2 Fit FNF2

−30 −32 −34

6

8

10 12 14 1000/V [Volts]

16

18

FIGURE 53. The data of Figure 52 represented on a Fowler–Nordheim Plot.

TABLE 7 FOWLER‐NORDHEIM FACTORS Curve

Slope (Exp)

Intercept (Exp)

˚] a [A

Slope (Theory)

Intercept (Theory)

I1 I2 F1 F2

728.73 1331.0 1580.0 1623.7


17.385
18.340
17.659
17.560

42.9 93.8 118.6 123.1

728.44 1331.8 1580.0 1623.3


17.85
18.10
18.14
18.14

Charbonnier, 1998), so that tips conditioned in such a manner can be made more like each other, thereby improving emission uniformity from an array of such emitters. Our purpose here, however, is to determine whether such changes are captured in the variation of the theoretical model of the slope and intercept factors. The curves labeled ‘‘I1’’ and ‘‘I2’’ are as fabricated, whereas the curves labeled ‘‘F1’’ and ‘‘F2’’ (following the notation of Schwoebel et al.) are after conditioning. Table 7 shows the slope and field values, from which the effective radius is determined for a presumed average work function of 4.5 eV. As opposed to inferring both field enhancement factor and work function from slope‐intercept data, the work function here is held at a presumed value, the slope is used to infer the apex radius based on a field enhancement model, and the intercept is predicted. For aforementioned reasons, the intercept should not be expected to be exactly predicted (see, for example, Forbes, 1999b for a general discussion on the problems associated with inferring emission area and work function from FN

131

ELECTRON EMISSION PHYSICS

slope‐intercept data). As seen in the table, the general trend is captured and the data shown to be commensurate with the hypothesis that the tips are in fact blunting through conditioning to about the same magnitude. G. Recent Revisions of the Standard Thermal and Field Models 1. The Forbes Approach to the Evaluation of the Elliptical Integrals Up to this point, the methodologies used to tackle thermal and field emission in the pursuit of emission equations have not departed significantly in the reliance on common representations of the elliptical functions v(y) and t(y) to determine n(F,T) and the onset of the transition region. Recent improvements have enabled a truly general thermal‐field equation beyond the simple addition of correction terms used by the revised FN and RLD equations that moreover does not rely on numerical searches for the integrand maximum in the transition region; they make use of recent advances by Forbes (2006) in creating extraordinarily convenient and elegant analytical forms of v(y) and t(y) over the entire range of y. We first explore the development of the analytical forms and then apply them to the physics of the transition region of the thermal‐field model. Although the goal is the form given by Forbes, the path is different and based on series expansion methods. Recall the defining equation for v(y) introduced in Eq. (260), which facilitates the development of a series summation that will be particularly useful and rewrite it as

npffiffiffiffiffiffiffiffiffiffiffiffiffio 3 pffiffiffi vðyÞ ¼ 2 1
y2 S 1
y2 8 ðp ð324Þ sin2 y S fxg ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidy 1 þ xcosy 0 The slight rewriting has immediate payoff, as S can be series expanded to give 8 9 S ðxÞ ¼

¼

ð p=2 < = 1 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2 ydy 1
xcosy; 0 : 1 þ xcosy

X1

¼2

ð
1Þn n¼0

X1 n¼0

ð2nÞ! 2n

2 ðn!Þ

ð4nÞ! 24n ðð2nÞ!Þ2

ð p=2 xn 2

x2n

0

ð p=2 0



ð
1Þn sin2 ycosn y þ sin2 ycosn y



ð325Þ sin2 ycos2n ydy

1 X1 ð4nÞ! x2n ¼ p n¼0 6n 2 2 ð2nÞ!ðn!Þðn þ 1Þ!

132

KEVIN L. JENSEN

Initially, this appears to be of little benefit, but a great simplification arisespifffiffiffiffiffiffiffi the ffi lowest‐order approximation to n! (Stirling’s approximation, or n!  nn e
n 2pn) is used, for which S(x) becomes approximated by So(x) where (note the changes in the summation limits) p 1 pffiffiffi X1 x2n 2 þ n¼1 nðn þ 1Þ 2 4 pffiffiffi p 1 pffiffiffi 2ð1
x2 Þlnð1
x2 Þ 2þ ¼ þ 2 4 4x2

So ðxÞ 

ð326Þ

where the integration of the commonly known series expansion for ln(1–s) with s ¼ x2 has been exploited to convert the summation into a closed formula. Using So in place of S, it follows that v(y) is approximated to leading order by pffiffiffi

3 3 1 þ p 2 1
y2 þ y2 lnðyÞ: ð327Þ v ð yÞ  16 8 As simple as the result appears, it cannot be correct; while v(1) ¼ 0 (as it should), v(0) does not equal 1, but rather 1.0205, and the problem is traceable to the ungainly coefficient of ð1
y2 Þ. The consequences of using Stirling’s approximation have made themselves felt. It was Forbes’ insight, based on examining the tabulated function and then experimenting on expansions and summations using the Maple mathematical package (Maplesoft, Waterloo, Ontario, Canada), that the better approximation is

1 vðyÞ  1
y2 þ y2 lnðyÞ; 3

ð328Þ

where the coefficient (1/3) was ascertained to be a fairly close fractional representation of the actual numerical coefficient. The elegant simplicity of Eq. (328) is breathtaking for those who have squandered countless hours searching for a good analytical representation: it contains the proper end points and at its worst is still good to within 0.332% of the numerically evaluated value (occurring at y ¼ 0.54). Moreover, the form lends itself to the ready evaluation of t(y) and the interpretation of FN slope factors. A little effort shows why the (1/3) coefficient is in fact a reasonably good approximation, how good ‘‘good’’ is, and how close Eq. (328) is to a proper account of the summation terms. Compared to the ‘‘derivation’’ of Eq. (328), it is not as appealing and relies on some patience with series expansion methods (Jensen, 2007). Introduce a difference function D(x) that varies from 0 to 1 and represents the difference between S(x) and its approximation So(x). Using the series form of each, it follows that

ELECTRON EMISSION PHYSICS

DðxÞ 

ðSðxÞ
So ðxÞÞ
ðSð0Þ
So ð0ÞÞ SðxÞ
So ðxÞ pffiffiffi ¼ 12 ; ðSð1Þ
So ð1ÞÞ
ðSð0Þ
So ð0ÞÞ 13 2
6p

133

ð329Þ

so that 

pffiffiffiffiffiffiffiffiffiffiffiffiffi 3 pffiffiffi  1 pffiffiffi 2 2 2x So ðxÞ þ 13 2
6p DðxÞ : v 1
x ¼ 8 12

ð330Þ

It is seen that Eq. (327) results if the term containing D(x) is neglected. The term So(x) has explicitly selected the singular term for vanishing x. Therefore, whatever remains is a rapidly convergent power series in x2, or equivalently, in y2, that is v

pffiffiffiffiffiffiffiffiffiffiffiffiffi X1  2n



3 1
x2 ln 1
x2 1
x2 ¼ A x þ n¼1 n 16 n X1 2 n o 3 2 vðyÞ ¼ ð1
y2 Þ 1 þ A
y þ y lnðyÞ n n¼1 8

ð331Þ

where the first summation in the top representation starts at n ¼ 1 because v(1) ¼ 0, and the bottom term in the curly brackets is a consequence of both regrouping (the reason for the asterisk in the x power series being dropped in the y power series) and the demand that v(0) ¼ 1; it is precisely that observation that allows us to bypass the problematic boundary condition at y ¼ 0 if we choose to truncate the series expansion after a few terms, which is our intention to obtain the An coefficients of the y‐power series. Therefore, consider the first handful of terms in the series expansions of v(y) and So(x) from Eqs. (324)–(326), that is 0 1 pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi

3 3 35 x4 A v 1
x2  p 2x2 @1 þ x2 þ 16 32 1024 0 1 p 1 pffiffiffi 2 @ 1 2 1 4A 2x 1 þ x þ x So ðxÞ  þ 2 8 3 6

ð332Þ

and insert these into Eq. (330) to determine D(x): pffiffiffi pffiffiffi  pffiffiffi



 2x2 96 3 2p
16 þ 105 2p
512 x2 þ Oðx4 Þ DðxÞ  : pffiffiffi

1024 13 2
6p

ð333Þ

134

KEVIN L. JENSEN

Clearly, truncating the series early incurs an error that is increasingly large when x approaches unity. The affront to a finite series representation of D(x) is minimized, therefore, by appending a correction term only to the coefficient of the highest power kept. That is, if f(x) is an infinite series and fa(x) is a finite series approximation to it, or f ðx Þ ¼ fa ðxÞ ¼

X1 k¼1

ak xk

Xn

a xk þ anþ1 xnþ1 k¼1 k

ð334Þ

and where both vanish at x ¼ 0 and are unity at x ¼ 1, then anþ1 ¼

X1 j¼nþ1

aj ¼ 1


Xn j¼1

aj :

ð335Þ

The error of the approximation vanishes at the boundaries and is f ðxÞ
fa ðxÞ ¼
xnþ1

X1 j¼1



anþjþ1 1
xj

ð336Þ

in the middle. Demanding that D(1) ¼ 1 determines the correction to the last coefficient, and so, using Eqs. (334) and (335) DðxÞ )

pffiffiffi pffiffiffi pffiffiffi  pffiffiffi





 2x2 96 3 2p
16 þ 105 2p
512 x2 þ 15 512 2p
512 x4 pffiffiffi

1024 13 2
6p

ð337Þ Putting Eq. (337) plus the closed form of Eq. (326) into Eq. (330) and then collecting terms shows that if the series is truncated at n ¼ 3 in Eq. (331), then 9897 pffiffiffi 85 p 2
 0:02754 16384 32 5145 pffiffiffi 89 A2 ¼ p 2
 0:009114 8192 32 15 pffiffiffi 15 A3 ¼ p 2
 0:0021112 16384 16 A1 ¼

ð338Þ

Using these values, it follows that truncating the series at the third term gives rise to the approximations

ELECTRON EMISSION PHYSICS



   3 vðyÞ  y2 lnðyÞ þ 1
y2 1 þ y2 A1
A2 y2 þ A3 y4 8 8 9 < = 1 2 13 tðyÞ 
y lnðyÞ þ 1 þ y2 B1 þ B2 y2
B3 y4 þ A3 y6 : ; 8 3

135

ð339Þ

where 3299 pffiffiffi 31 p 2þ  0:074153 16384 32 33645 pffiffiffi 145 B2 ¼ p 2
 0:061084 16384 16 41265 pffiffiffi 357 B3 ¼ p 2
 0:033665 16384 32

B1 ¼


ð340Þ

and where the maximum error of Eq. (339) is 0.029% for v(y) and 0.039% for t(y). Using this representation, which is designed to be accurate at the boundaries, it can be shown that the boundaries are (correctly) given by vð0Þ 0 ¼ tð0Þ 1¼1 vðyÞ A 3 pffiffiffi lim @ ¼ p 2 y!1 1
y2 16 1 pffiffiffi tð1Þ ¼ p 2 4

ð341Þ

It must be emphasized that the use of a truncated series in An* to find a truncated series in An is a procedure that does not provide the exact values of An (even though for all but AN*, the An* coefficients are specified exactly) but rather approximations to these terms. However, approximations are all we are seeking; namely, we are striving to find a method that steadily improves commensurate with the level of effort corresponding to the number of terms retained. How does this relate to Forbes’ beautiful result? We seek to confirm the value of C in

vðyÞ  1
y2 þ Cy2 lnðyÞ

ð342Þ

such that the y ¼ 1 limit of Eq. (331) is reproduced in Eq. (342) (field emission conditions are such that y is generally closer to that boundary). We find

136

KEVIN L. JENSEN

8 <3

9 =  ð1
y2 Þ 
A 1
A 2 y2 þ A 3 y4 C ¼ lim ; y!1 :8 lnðyÞ 3 pffiffiffi ¼ p 2 þ 2  0:33392 8

ð343Þ

where, from L’Hoˆpital’s rule, the y ¼ 1 limit of ð1
y2 Þ=lnðyÞ ¼
2. Therefore, the finding that C be approximated by 1/3 is well supported. It follows that 2 tðyÞ ¼ vðyÞ
y]y vðyÞ 3 1 2 1 5 
y lnðyÞ þ 1 þ ð1
4A1 Þy2 þ ðA1 þ A2 Þy4 8 12 3 13 A 3 y8 3 1 1 
Cy2 lnðyÞ þ 1 þ ð1
2C Þy2 3 3
3ðA2 þ A3 Þy6 þ

ð344Þ

where the second line uses the power series (An) approximation and the third uses the Forbes‐like (C) approximation. Finally, the choice of C defined by Eq. (343) entails that the boundaries in Eq. (341) are respected. Figure 54 shows the performance of both the An and C forms of v(y) and t(y). Finally, note that what is designated C is actually C(y¼1). The Forbes‐like equations can be retained with theirpevident ffiffiffiffiffiffiffiffiffiffiffi simplicity and utility by replacing C(1) with C(yo), where yo ¼ 4QFo =F and Fo is a characteristic or midpoint temperature for the experiments under consideration. The behavior of C(y) is shown in Figure 55. However, given that the Forbes approximation is better than 0.4% for all y, the incentive to do so is rather weak. 2. Emission in the Thermal‐Field Transition Region Revisited The punishing analysis to obtain the deceptively simple Forbes‐like representation of the elliptical integral functions v(y) and t(y) serves a purpose: it is precisely in the y ¼ 1 limit that the transition region from field to thermal emission occurs. We may now profitably revisit the n ¼ bT/bF approximations to the thermal‐field equation and deal with the transition region explicitly. The behavior of the transition region was the subject of a detailed investigation by Dolan and Dyke (1954, in their Figure 2), which chronicles the evolution of the transition region as a function of both temperature and

137

ELECTRON EMISSION PHYSICS

0.4

% error

0.3

0.2

% Error v(A) % Error v(C) % Error t(A) % Error t(C)

0.1

0

0

0.2

0.4

0.6

0.8

1

y FIGURE 54. Comparison of the Forbes approximation (C = 1/3) with the polynomial approximation to C(y) in Eq. (342)

0.37 Cquad(y) = 0.368 − 0.049y + 0.015y2

C(y)

0.36 C(y) Quad C(1)

0.35

0.34

0.33

0

0.2

0.4

0.6 0.8 1 y FIGURE 55. Comparison of the numerical, polynomial fit, and constant (1/3) representations of C(y).

field, albeit using the FN factors; the intent here is to improve upon the analyses. Recall the compelling behavior in Figure 40: the transition from thermal to field emission‐like regimes is demarcated by a regime in which n is unity—though not exactly, still to a very good approximation. This suggests the FN slope factor bF (which depends on v(y)) can be used up to the transition region, and the quadratic barrier slope factor can be used after the transition region. What is needed now is an analytic method to find the slope factor within the transition region for which n ¼ 1 (recall that the numerical method is to simply crudely sum the current density integral using any appropriate numerical algorithm after finding the slope factor bF

138

KEVIN L. JENSEN

and expansion point Eo using numerical search algorithms). What is happening while n ¼ 1 is that the peak of the current density integrand as a function of energy is migrating from the Fermi level to the barrier maximum level (see, for example, Figure 8 of Murphy and Good, 1956), and in that simple observation, a solution is suggested. The expansion of AUC term y(E) constitutes the starting point, where it was observed that in the transition region between the FN and quadratic barrier forms, the slope factor bF appeared to be well approximated by a quadratic in energy. Using a standardized notation, a polynomial approximation in the difference between energy and Fermi level for y(mþxf) of the form ya ðm þ xfÞ  Ba þ Ca x þ Da x2 þ Ea x3

ð345Þ

is sought in such a way that the two linear approximations of primary interest, namely, the linear FN given by yfn ðm þ xfÞ ¼ BFN
CFN fx 4 pffiffiffiffiffiffiffiffiffiffiffi3ffi BFN ¼ 2mF vðyÞ 3
hF 4f pffiffiffiffiffiffiffiffiffiffiffi3ffi CFN ¼ 2mF tðyÞ
hF

ð346Þ

and quadratic barrier approximations given by yq ðm þ xfÞ ¼ Bq
Cq x 0 11=4 p pffiffiffiffiffiffiffi@ Q A Bq ¼ Cq ¼ f 2m h
F3

ð347Þ

pffiffiffiffiffiffiffiffiffiffi where y ¼ 4QF =F and f ¼ ð1
yÞF, shall determine the coefficients in Eq. (345). Note explicitly that x is dimensionless, its value at 0 designating an energy at the chemical potential, and its value at 1 designating the barrier maximum. Demand that y(E) be approximated by the fn‐form for E < m, by the q‐form for E > m þ F, and by ya in the intermediate region, where the value of ya and its first derivative are continuous at the boundaries with the linear forms. We find for x ¼ ðE
mÞ=f that ya ðm þ xfÞ ¼ BFN
CFN x þ x2





 CFN
Cq ð2
xÞ
BFN
Bq ð3
2xÞ

ð348Þ

ELECTRON EMISSION PHYSICS

139

and for the slope factor

] y ðE Þ ]E

 1 Bq x þ CFN ð1
xÞ þ 3 2BFN
Bq
CFN xð1
xÞ  f

b F ðE Þ 


ð349Þ The transition region approximation shall be invoked when the integrand maximum expansion point lies between m and m þ f, at which point n will be taken as identically 1 and the integrand expansion point x ¼ xc determined the condition bF ðE ðxc ÞÞ ¼ bT (another way of saying n is identically equal to unity) and for which 0 1 2F C o pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA Eo ¼ m þ @ Bo þ B2o þ 4Ao Co Ao ¼ 6BFN
3Bq
3CFN

ð350Þ

Bo ¼
6BFN þ 2Bq þ 4CFN Co ¼
bt F þ CFN As an example, consider copperlike parameters (m ¼ 7.0 eV, F ¼ 4.5 eV) under conditions of a field of 1 GV/m and a temperature of 723 K, and assuming the Forbes approximation for v(y) and t(y): then Ec – m ¼ 2.27 eV, or xc ¼ 0.687. A key limitation of the revised FN and RLD equations has now been overcome, namely, the specification of the Eo parameter by other than numerical means. We are now in a position of evaluating bF and Eo without relying solely on the FN linear approximation which, for mixed thermal‐field conditions, was unsatisfactory. We therefore turn to the development of a truly general thermal‐field equation. H. The General Thermal‐Field Equation As shown by Figure 40, to a good approximation, n ¼ 1 in the transition region between the thermal and field regimes. A reasonable approximation can then be made by taking n to be equal to 1 when the temperature falls within a critical region that occurs when T is larger than the FN‐like temperature TFN yet smaller than the RLD‐like temperature TRLD, both of which are obtained by finding the equivalent temperatures corresponding to the slope factors. In other words, n ¼ 1 when

140

KEVIN L. JENSEN

TFN 

1 1 T   TRLD : kB bF ðmÞ kB bF ðm þ fÞ

ð351Þ

For completeness, when temperatures are above TRLD or below TFN, then Eo ðT > TRLD Þ ¼ m þ f 2vðyÞ F Eo ðT < TFN Þ ¼ m þ 3tðyÞ

ð352Þ

as before, but when bF ðEm Þ  bT (a relation that serves to define Em in the transition region), then Eo  Em þ

yðEm Þ ; bF ðEm Þ

ð353Þ

where the boundary cases for field (Em ¼ m) and thermal (Em ¼ m þ f) in Eq. (353) agree with Eq. (352). In turn, s(Em) is always given by sðF ; T Þ  bF ðEm ÞfEo ðEm Þ
mg

ð354Þ

from which it can be shown that s(TRLD) ¼ Bq and s(TFN) ¼ BFN. The remaining terms still require a tractable form in order to obtain a truly general thermal‐field equation. Starting from the form of N(n,s)   1
s N ðn; sÞ ¼ S ð355Þ e þ SðnÞe
ns n [where the expressions of J(F,T), JT, and JF have not changed from their forms given in Eq. (299)], recall that S(x) can be written as X1

SðxÞ ¼ 1 þ 1
21
2j zð2j Þx2j : ð356Þ j¼1 Such a form can be cumbersome; a reasonable approximation is given by SðxÞ 



1 1
xð1 þ xÞ þ x3 ð7x
3Þ þ zð2Þx2 1
x2 ; 1
x 4

ð357Þ

where, as done above on other infinite series, the highest‐order term is amended by the next‐order term to respect boundary conditions (in the present case, making the nonsingular part of S(1) unity). The singular parts of Eq. (355) at n ¼ 1 cancels, and the remaining terms are well behaved. Finally, the Forbes approximation is used to revisit the power law dependence of n on F, which can be rewritten as  p F nðF Þ ¼ nðFo Þ ; ð358Þ Fo

ELECTRON EMISSION PHYSICS

141

where Fo is a reference field and p is a power. As shown in Eq. (307), ptherm ¼ 3/4. It is the form of pfield that was numerically found in Eq. (308) but for which the Forbes approximation now allows an analytical expression to be ascertained. Introduce y2o ¼ 4QFo =F2 . It then follows from Eq. (358), the field‐regime term bF(m), constant temperature, and the Forbes expression for t(y) that for F close to Fo we discover pfield ¼

18 þ y2o 18 þ y2o ¼ : 18tðyo Þ 18 þ 2y2o ð1
lnðyo ÞÞ

ð359Þ

Unfortunately, Eq. (359) continues to rely on a reference field Fo. However, a graph of pfield exhibits a minimum at yo ¼ 0.9740 for which pfield ¼ 0.950, close to the numerical value discovered previously. In practice, pfield ¼ 0.950 remains a good approximation to p for n > 1. To summarize the approximation, the behavior implied by increasing the field monotonically from low values to high is that initially, the location of the current density integrand maximum is at the barrier maximum as F increases and n increases. When n ! 1
(the temperature is equal to TRLD), the location of the integrand maximum begins to migrate from the barrier maximum to the chemical potential, caused by the smooth change of the field slope factor from quadratic barrier–like to FN‐like. When n ! 1þ , then the location of the current density integrand maximum takes root at the chemical potential and n increases to larger values. Figure 56a shows the performance for copperlike parameters with the temperature held at cold (300 K) or hot (1500 K) conditions, indicating that the RLD and FN approximations work rather well in their respective regimes. The performance in the intermediate regime is obfuscated on a log‐log plot, so Figure 56b shows the ratio of the FN and RLD currents with the thermal‐field model, showing how well each equation performs in the transition regime. Switching conditions to cesium on tungsten cathode‐like conditions but for intermediate temperatures and fields (and in particular, for a work function of 2.0 eV, slightly higher than the 1.8 eV suggested earlier, simply for effect), the ratio comparison in Figure 56c shows the degree to which the FN and RLD models depart. The resultant general thermal‐field emission equation for which the two equations FN and RLD are shown to be limiting cases has been constructed and works for arbitrary n, even in the transition region specified by n ¼1. By formulating the theory in this manner, the present formulation allows for the determination of the effects of temperature on field emission as well as fields on thermal emission. The ability to unify the equations was a consequence of developing a good approximation to y(E) that smoothly transitioned from below the barrier to above it. We shall see below that the example of photoemission benefits by extending the analysis begun here.

142

KEVIN L. JENSEN

log10(J [A/cm2])

(a)

8

Cu-like m = 7 eV, Φ = 4.5 eV

4

JTF(300 K) JTF(1500 K)

0

JRLD(1500 K) JFN

−4 −8 −3

−2.5

−2 −1.5 −1 log10(field [eV/Å])

−0.5

0

Ratio of current

(b) 10 Cu-like 1500 K m = 7 eV, Φ = 4.5 eV

1 JRLD/JTF JFN/JTF 300 K

R=1 0.1 0.001

0.01 0.1 Field [eV/nm]

1

(c) 100 300 K

JTF/JRLD Ratio of current

JTF/JFN R=1

10

1500 K 1 Cs-on-Cu-like m = 7 eV, Φ= 2 eV 0.1

0.02

0.04

0.06

0.08

0.1

Field [eV/nm] FIGURE 56. (a) Comparison of the thermal-field equation (JTF) with both the Richardson– Laue–Dushman equation (JRLD) and the Fowler–Nordheim equation (JFN) for copperlike conditions for 300 K and 1500 K.. (b) Ratio of the thermal-Field equation (JTF) with both the

ELECTRON EMISSION PHYSICS

143

I. Thermal Emittance Consider a symmetrical beam of electrons and let the symmetry axis be ^z. If all electrons had their velocities wholly in the direction of the symmetry axis (that is, if k ¼ k^z), then the beam would not diverge as it propagates. Electrons, however, are always emitted on average with some perpendicular velocity component kr , and as the beam moves in the forward direction, these electrons find themselves farther and farther away from the symmetry axis. If no forces complicate matters, then the ratio of the spread of the beam 0 to the distance the beam has propagated goes as x ¼ dx=dz  dkx =dkz , where the last relationship assumes that the axial velocity dominates the 0 radial velocity. If every particle was tagged by a pair of coordinates ðx; x Þ 0 and these points plotted on the axes x and x , then the area that encompassed all of the points—that is, the trace space defined by ðð Ax 

dxdx

0

ð360Þ

provides a measure of the quality of the beam. Problems inherent with a trace‐space definition of ‘‘emittance’’ are discussed more fully by Reiser (1994), even though the quantity is commonplace in the literature, but for ideal beams with linear focusing fields, the relationship between the rms emittance ~ex (rms ¼ root‐mean‐squared and relates to the statistics of the distribution of points; see below) and the trace space of Eq. (361) is Ax ¼ 4pe x :

ð361Þ

As seen in Eq. (360), the units of emittance are a bit odd on first encounter; while x has units of length, x0 does not—rather, it has units of radians. In the community of electron sources, a commonly used unit is 10
6 meter‐radians, or, as it is more often encountered, mm‐mrad. Although ‘‘microns’’ are also used, such a designation obscures the angular nature inherent in the definition of emittance. Another measure of the quality of a beam is the total beam current for a given emittance, which can be shown to be related to the current density for a given solid angle. Brightness is therefore defined as Richardson–Laue–Dushman equation (JRLD) and the Fowler–Nordheim equation (JFN) for copperlike conditions. (c) Ratio of the thermal-field equation (JTF) with both the Richardson– Laue–Dushman equation (JRLD) and the Fowler–Nordheim equation (JFN) for cesium on copperlike conditions.

144

KEVIN L. JENSEN

B ¼ J=dO:

ð362Þ

For idealized particle distributions whose trace‐space is confined by a hyperellipsoid (an ellipsoid in four dimensions (x,y,x0 ,y0 )), it can be shown that the average brightness is given by hBi ¼

2I p2 ex ey

ð363Þ

and therefore has the units of A/(mm‐mrad)2. As seen previously, the distribution function approach leads naturally to a continuity equation, ]t r þ r J ¼ 0. Generalizing, if the current is represented as the product of a (six‐dimensional) density in phase space and a velocity, then





]t r þ v rr ¼

dr ¼0 dt

ð364Þ

if the number of particles dN in a small region dV is not changing, then dN ¼ rdV (again, it is emphasized that dV is a small volume in phase space and therefore six‐dimensional). Thus, 0 1 0 1 d dr d dN ¼ @ AdV þ r@ dV A dt dt dt 0 1 ð365Þ d ¼ r@ dV A ¼ 0 dt where the second line follows as a consequence of Eq. (364). We conclude ðð d d dV ¼ dxdk ¼ 0: ð366Þ dt dt That is, the volume of a given number of particles in phase space is invariant, a conclusion known as Liouville’s theorem (see Reiser, 1994, for a discussion). Insofar as coupling does not occur between motion in the various directions, the finding of the invariance of the phase space volume is equivalent to invariance of its projections on various pairs of axes such as dxdkx, and so it is found that trace‐space area is conserved. By extension, this has bearing on the behavior of the emittance as per the relationship between Ax and ex. Cathodes for advanced accelerators and advanced linear accelerator (LINAC)‐based light sources, vacuum electronic devices, high‐energy physics, and the like are responsible for generating well‐collimated beams as the consequences of errant electrons outside the intended path lead to very

ELECTRON EMISSION PHYSICS

145

undesirable results (Bohn and Sideris, 2003; O’Shea, 1995)—stray electrons from a high‐energy beam still have a strong negative impact on whatever they strike. Intrinsic emittance, that is, emittance originating with the photocathode, is important because what is generated there cannot be compensated for by subsequent beam optics. Emittance e appears in the envelope equation (Reiser, 1994; Serafini and Rosenzweig, 1997) as a parameter governing the evolution of the beam radius (r)   2 I 1 e2 00 2 r þ ko r

¼ 0; ð367Þ r Io ðbgÞ3 r3 where betatron wave number of the focusing fields, b ¼ vz =c, and ko is the 1=2 g ¼ 1
b2 are the dimensionless velocity and relativistic correction factor, Io ¼ 4pe0 mc3 =q ¼ 17045 A is a characteristic current. Emittance is related to the ‘‘moments’’ hx2 i and hx02 i, where x0 ¼ dkx =dkz where
hk=m is the velocity of the particle, kx being the conjugate variable to x. The related rms emittance is defined by erms ¼ e/4 for a uniform beam. A beam without emittance may propagate with pencil‐like straightness, whereas when emittance is present, the beam can diverge and the extent to which it diverges over a given propagation distance is a measure of the transverse velocity components. Brightness is also affected, and so a normalized brightness is also defined in terms of the normalized emittance as Bn 

B ðbgÞ

2

¼

2I : p2 e2n

ð368Þ

When the particle velocity is small or when the transverse and perpendicular components are comparable (i.e., near the cathode), then using x0 poses problems so the definition used here is (O’Shea, 1998) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h en;rms ðzÞ ¼ hx2 ihk2x i
hxkx i2 : ð369Þ mc Moments are defined according to ð drdkOðr; kÞf ðr; kÞ ð hOi  drdk f ðr; kÞ;

ð370Þ

where f is the distribution of emitted particles. Attention shall be restricted to axisymmetric beams for which hxkx i ¼ 0. In the case of thermionic emission, only those electrons whose energy exceeds the barrier height (m þ f) may be emitted and so

146

KEVIN L. JENSEN

f ðr; kÞ ¼ yðr
rc ÞyðEðkz Þ
m
FÞfFD ðEðkÞÞ;

ð371Þ

h2 k2z =2m, rc is the radius of the cathode where EðkÞ ¼
h2 jkj2 =2m, Eðkz Þ ¼
(cylindrical coordinates), y is the Heaviside step function, and fFD is the FD distribution. The symmetry of the distribution results in hx2 i ¼ hr2 i=2 ¼ r2c =2. For typical work functions, the distribution function for energies above the barrier height is well approximated by a MB distribution, and so the moment for momentum is equally straightforward. Minimal effort shows that ð1 n o exp
bT
h2 k2r =2m k3r dkr m hk2x i ¼ hk2r i=2  ð01 ¼ ; ð372Þ n o 2 2 exp
bT
h2 k2r =2m kr dkr bT
h 0

yielding the oft‐quoted result that the emittance of a thermionic cathode is rc en;rms ðthermalÞ ¼ : ð373Þ ð4bT mc2 Þ1=2 A numerical example is to consider a cathode 0.5 cm in radius and at 1300 K. Eq. (373) then indicates that the emittance is 1.171 mm‐mrad. Two points merit attention, as the question of emittance is considered afresh in the treatment of photoemission in the effort to derive an equation of comparable simplicity to Eq. (373). First, if ‘‘moments’’ of the distribution function are defined by ð Mn / knr f ðEðkÞÞdk; ð374Þ then Eq. (372) proportional to M2 =2M0 . Second, the replacement of the FD by the MB distribution is crucial to facilitate the stunning ease by which Eq. (372) is obtained. In a more general circumstance, such as in photoemission, the convenience entailed by the MB distribution will be of no avail. Conversely, emittance for field emission is so significantly complicated by questions of field variation over sharpened emitter structures and the change of field lines with emitted charge that the evaluation of emittance for such structures is a question of considerable complexity (Jensen et al., 1996, 1997) and is not considered further here.

ELECTRON EMISSION PHYSICS

147

III. PHOTOEMISSION A. Background The explanation of the photoelectric effect in terms of quanta liberating electrons from the surface of a metal earned Albert Einstein the Nobel Prize in 1921. As interesting as the liberation of a few electrons is, the liberation of many electrons complicates the physics significantly and affects the transition of that physics into technology. The approach in this section continues its focus on electron emission and current density; thus the treatment of photoemission is considered in that light. Photocathodes are excellent sources for the production of short bunches of electron beams for injection into radiofrequency (RF) LINACs, free electron lasers, and related devices (Nation et al., 1999; Rao et al., 2006). While requirements vary, the European Organization for Nuclear Research (CERN) (linear collider) test accelerator is a measure of the state of the art. It uses a Cs2Te photocathode illuminated with a 262‐nm yttrium‐lithium‐ fluoride (YLF) laser to generate electron bunches postaccelerated to 50 MeV containing 30 nC per bunch at a modulation frequency of 3 GHz. As such, its nominal characteristics are in interesting contrast to field and thermionic technologies; the fields at the surface are an order of magnitude greater than thermionic sources but two orders smaller than field emission sources, yet its average current is 10 A and the peak current substantially higher. Other photocathodes in use at, for example, at the Stanford Linear Accelerator (SLAC), Thomas Jefferson Lab National Accelerator Facility (JLAB), the ELETTRA Synchrotron Light Source in Trieste, and the German Electron Synchrotron (DESY), make related demands, although the details differ depending on the circumstance (mostly in charge per bunch and repetition rate). What is demanded of photocathodes modifies what merits discussion: demand much, and interesting physics is thereby revealed. Representative numbers that drive much of the following text (but do not represent a realized achievement) are 1 nC per bunch produced in 10 ps every 1 ns from a 1‐cm2 area corresponds to peak current density of 1 kA/cm2, and an average current density of 1 A/cm2. If such numbers were realized, then a megawatt (MW) class free electron laser (FEL) would be potentially brought to realization, so these are, in fact, numbers of interest. Intense current densities from sub–square‐centimeter regions are not uncommon, so much so that space charge effects within the bunch can affect its dynamics in nontrivial ways (Dowell et al., 1997; Harris, Neumann, and O’Shea, 2006). For comparison, using typical numbers suggested by Dowell et al. for a current and current density of 77 A and 530 A/cm2, respectively, a

148

KEVIN L. JENSEN

pancake bunch containing 2 nC from an area 0.145 cm2 produces a local field of approximately (2 nC)/(0.145 cm2)2e0 ¼ 7.8 MV/m, which is sufficient to affect internal structure and adjacent bunches. At extraordinarily high laser intensities, multiphoton effects are revealed, wherein the quantum efficiency depends on higher powers of laser intensity than simply a linear relation; further, the electron gas can be brought to such temperatures so quickly that thermionic emission results even as the electron gas temperature decouples from that of the lattice (Girardeau‐Montaut et al., 1993, 1994, 1996; Girardeau‐Montaut and Girardeau‐Montaut, 1995; Logothetis and Hartman, 1969; Papadogiannis and Moustaizis, 2001; Papadogiannis, Moustaizis, and Girardeau‐Montaut, 1997; Papadogiannis, Moustaizis, Loukakos, and Kalpouzos, 1997; Papadogiannis et al., 2002; Riffe, Wertheim, and Citrin, 1990; Riffe et al., 1993; Tomas, Vinet, and Girardeau‐Montaut, 1999).

B. Quantum Efficiency The ability to liberate electrons for a given laser intensity is measured by quantum efficiency (QE). Various materials commend themselves for different reasons. Metal photocathodes are rugged and prompt emitters and can produce very short bunches but require higher‐intensity lasers to do so as their QEs are on the order of 0.001–0.01% (Papadogiannis, Moustaizis, and Girardeau‐Montaut, 1997; Srinivasan‐Rao, Fischer, and Tsang, 1991, 1995). Semiconductor photocathodes such as GaAs require much lower‐intensity drive lasers and can produce polarized electron bunches, but they generally require better vacuum conditions since they are more fragile (Aleksandrov et al., 1995; Maruyama et al., 1989). Direct bandgap p‐type semiconductors (alkali antimonides and alkali tellurides; Michelato, 1997; Spicer, 1958; Spicer and Herrera‐Gomez, 1993), and bulk III‐V with cesium and oxidant (Maruyama et al., 1989) have high QEs on the order of 30% but are chemically reactive and easily poisoned, damaged by back ion bombardment (Sinclair, 1999), and for Negative Electron Affinity (NEA) III‐V photocathodes, which have excellent QE, have a long response time of tens of picoseconds (Table 8). The required drive laser intensity is related to the number of electrons that can be liberated for a given number of incident photons on the surface of a material. (The speed with which lasers can be turned on and off coupled with a fast‐response photocathode enables the generation of bunches of electrons with a short spatial extent that is unavailable by other means, thereby explaining the strong interest of the technology, for example, in the accelerator community when RF photoinjectors are used; Michelato, 1997; O’Shea et al., 1993; Travier, 1994). An electron absorbing a photon will be raised in energy by an amount
ho. If
ho > F, then the electron has a nontrivial

Material

n

l (nm)

Efficiency [%]

QE (%)

Lifetime

Time response

K2CsSb Cs2Te GaAs Cu Mg Goal

2 4 2 4 4 3

532 266 532 266 266 355

50 10 50 10 10 30

8 5 5 1.4
102 6.2
102 1

4 hours >100 hours 58 hours >1 year >1 year kHr 0

Prompt Prompt <40 ps Prompt Prompt Prompt

Vacuum tolerance

Power at photo‐cathode (W/cm2)

1.06‐mm drive laser power (W/cm2)

Poor Very good Poor Excellent Excellent Excellent

44 140 70 50043 11300 525

88 1401 140 500430 113000 1752

ELECTRON EMISSION PHYSICS

TABLE 8 PHOTOCATHODE‐DRIVE LASER COMBINATIONS*

*Power on the photocathode at a specified drive laser frequency required to produce 1 nC from a photocathode area of 0.125 cm2 in 50 ps.

149

150

KEVIN L. JENSEN

probability of escape. The number of photons is the ratio of the amount of incident energy with the photon energy, or No ¼ DE=
ho, whereas the number of electrons is the ratio of the emitted charge with the electron unit charge, or Ne ¼ DQ=q. The QE is then the ratio of the number of emitted electrons with the number of incident photons, or
 Ne
ho DQ QE  ¼ : ð375Þ q DE No Metal photocathodes, being prompt, produce electron pulses that follow the light pulse, and the emission area is equal (or nearly so) to the illumination area. It follows that DE ¼ Io ADt and DQ ¼ Je ADt, where Io and Je are the illumination intensity of the incident light and the resulting current density, respectively, and A and Dt are the characteristic areas and pulse times. If the incident intensity is measured in watts per square centimeter and the current density in amps per square centimeter, then a convenient relation is
 ho Je
Je ½A=cm2  ð376Þ QE ) ¼ 1:2398 q Io l½mmIo ½W=cm2  The photon energy must exceed the work function, implying that ultraviolet (UV) light is needed (e.g., the fourth harmonic of a neodymium‐yttrium‐ aluminum‐garnet [Nd:YAG] laser, for which l ¼ 266 nm, corresponds to
ho ¼ 4:661eVÞ. The laser intensity required to obtain 1 A/cm2 from a photocathode with QE ¼ 0.01% is then 46.61 kW/cm2. A variety of issues are associated with such intensities. The ‘‘interesting physics’’ suggested by such conditions is twofold: (1) can the QE be predicted (and how), and (2) what impact do large laser intensities have? Regarding the prediction of QE, Spicer suggested a three‐step model (Berglund and Spicer, 1964b; Spicer, 1960; Spicer and Herrera‐Gomez, 1993) based on three events: (1) photon penetration and absorption, (2) electron excitation and transport to the surface, and (3) electron emission over the surface barrier. Spicer’s focus was principally on semiconductors (later compression of the electron beam can generate short pulses so that the higher QE of semiconductors can be profitably exploited), whereas the present discussion is on metals; the paradigm nevertheless is still useful. Until the discussion of the moments‐based formulation, the QE is therefore related to a product of factors—one accounting for absorption of light (leading to the treatment of reflectivity and penetration depth); one accounting for the probability of emission (leading to the notion of an escape cone, which will be replaced subsequently by a combination of the Fowler–Dubridge model and transport models); and one accounting for losses due to scattering during transport to the surface (leading to a model of a scattering loss factor).

ELECTRON EMISSION PHYSICS

151

C. The Probability of Emission 1. The Escape Cone Under the assumption that photoexcited electrons should be isotropically distributed, the QE then depends on what fraction of photoexcited electrons are optimally directed so as to surmount the surface barrier, thereby introducing the concept of an ‘‘escape cone’’ (mentioned here as such parlance appears in the literature), though other methods not beholden to the concept are adopted below. Fields that exist on the surface of a photocathode are typically on the order of 10–100 MV/m (for RF photoinjectors, which can support higher fields). As seen in the discussion of thermionic emission, fields of such magnitude preclude a tunneling contribution to the emitted current; therefore, it is sufficient to assume that the transmission probability is governed by the Richardson approximation, in which the electrons escape only if their momentum component directed at the surface (kx) is higher than the momentum corresponding to the barrier height, which, for a quadratic relationship between energy and momentum results in rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m kx > ðm þ fÞ  ko : ð377Þ h2
From the relation kx ¼ k cos(y), where y is the polar angle coordinate, the fraction fe of electrons that escape from the surface is given by (where Y is the Heaviside step function) Ð ð ð 2p 2 1 p O TÐ ðkx Þk dO ¼ sinðyÞdy dj Yðk cosðyÞ  ko Þ f e ðE Þ ¼ 2 4p 0 0 O k dO 0 0 1 sffiffiffiffiffiffiffiffiffiffiffiffi1 ð378Þ 1 ko 1 m þ fA ¼ @1  A ¼ @1  2 2 E k The integral over fe(E) (not to be confused with the supply function) is proportional to the QE, and for a zero‐temperature electron gas mþ
ðho

QE /

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ho þ fÞ  ðm þ fÞðm þ
hoÞ: fe ðE ÞdE ¼ m þ ð
2

ð379Þ

mþf

An expansion of Eq. (379) shows QE / ð
ho  fÞ2 . The fact that not all photoexcited electrons make it to the surface because their mean free path (distance between collision events) is less than their distance to the surface is considered separately when the impact of scattering is analyzed in greater detail.

152

KEVIN L. JENSEN

2. The Fowler–Dubridge Model The dependence of QE on photon energy and barrier height was uncovered by Fowler (1931), augmented by Dubridge (1933), and enjoys wide use (Bechtel, Smith, and Bloembergen, 1977; Girardeau‐Montaut and Girardeau‐Montaut, 1995; Jensen et al., 2003b; Jensen, Feldman, and O’Shea, 2005; Papadogiannis, Moustaizis, Loukakos, and Kalpouzos, 1997; Riffe et al., 1993). It can be easily understood in the context of the 1D supply function–transmission coefficient model familiar from the escape cone analysis, and as indicated in that model, it relies on the approximation that the effect of the photon energy is to raise Ex by an amount
ho (all the photon energy is directed at the surface). Such a conjecture, on the face of it, is overreaching, but as Fowler noted it is surprisingly effective in explaining experimental data and capturing its qualitative dependence, as follows from the limit of Eq. (379); for photon energies near the barrier height, unless Ex is augmented by the majority of the photon energy, the electron is unlikely to be emitted. The principal effect of augmenting Ex, then, is to make transmission more likely, so that T(E) in the current density integral is replaced with T ðE þ
hoÞ. The probability of emission is then a ratio of the current density emitted with the incident current density on the surface barrier. Electrons with an energy ho below the Fermi level are unlikely to find their final state unoccupied and
hence cannot make the transition. It follows that the probability of emission should then resemble Ð1 hoÞ f ðE ÞdE U ½bð
ho  fÞ m
ho T ðE þ
Ð1 ; ð380Þ Pð
ho Þ  ¼ U ½bm 0 f ðE ÞdE where the Richardson (thermionic) approximation to T(E) is used, and where the Fowler–Dubridge function U(x) has been introduced and is defined by Ðx U ðxÞ ¼ 1 lnð1 þ ey Þdy 1 ð381Þ ¼ x2 þ 2U ð0Þ  U ðxÞ 2 A special case is U(0) ¼ z(2) ¼ p2/12, where z is the Riemann zeta function. For negative argument, the log function can be series expanded to give U ðxÞ ¼

1 X ð1Þ jþ1 j¼1

j2

expðjxÞ;

ð382Þ

which is useful for large |x|. As observed in the treatment of the General Thermal Field Equation, for small |x| an approximate form good to better than 1% is U ðxÞ  ex ð1  beax Þ;

ð383Þ

153

ELECTRON EMISSION PHYSICS

where a and b are found by demanding that Eqs. (381) and (383) agree for U(x) and dU(x)/dx at x ¼ 0, or a ¼ ð1  lnð2ÞÞ=ð1  Uð0ÞÞ ¼ 1:7284 b ¼ ðUð0Þ  1Þ ¼ 0:17753

ð384Þ

The approximation given by Eq. (383) is shown in Figure 57, the relation for positive argument being trivially obtained by Eq. (381). To leading order, then, when the photon energy is in excess of the barrier height, the probability of escape becomes U½bð
ho  fÞ 6ð
ho  fÞ2 þ ðpkB T Þ2  : U ðbmÞ 6m2 þ ðpkB T Þ2

ð385Þ

As is often the case for metals under UV illumination, the difference between the photon energy and the barrier height term in Eq. (385) significantly exceeds the thermal term, and so the common observation that QE / ð
ho  fÞ2 results. When the photon energy, however, is comparable to the barrier height, then the thermal term makes its presence known (Figure 58), but clearly, the analytical approximation based on Eq. (385) is good for photon energies almost to the barrier height for moderate (e.g., room) temperatures and lower. For a metal like copper subject to a field of 10 MV/m and with incident 266‐nm laser light, the probability of emission suggested by Eq. (380) is 0.0714%, which is larger than reported values of QE for copper (Dowell et al., 2006; Srinivasan‐Rao, Fischer, and Tsang, 1991)— there is more physics in play, and we now turn to the other contributions.

U(−x)

100

U(x) Approx

10–1

10–2 0

1

2

3

4

5

x FIGURE 57. Comparison of the numerically calculated Fowler–Dubridge function with its analytical appoximation for negative argument [see Eq. (381) for positive argument].

154

KEVIN L. JENSEN

Probability of emission

10–2 Numerical Analytic Quadratic

10–3

10–4

Copper @ 500 K and 50 MV/m

10–5 260

270

280

290

Wavelength [nm] FIGURE 58. Comparison of the numerical evaluation of the Fowler–Dubrdige function with the analytic and quadratic approximations.

D. Reflection and Penetration Depth 1. Dielectric Constant, Index of Refraction, and Reflectivity The optical properties of solids are thoroughly discussed elsewhere (e.g., chapter 8 in Ziman, 1985, and chapter 6 in Marion and Heald, 1980). The present concern is with the degree to which light is reflected from a surface and the extent to which it penetrates into a metal. The electric field component E of an electromagnetic wave satisfies the propagating wave equation with dissipation derivable from Maxwell’s equations: =
E ¼ mo @t H =
H ¼ eo @t E þ J

(

 )


 ) 1 @ 2 1 @ J; =  2 E¼ c @t eo c2 @t 2

ð386Þ

where it is assumed that there is no spatial variation in electron density, eo and mo are the electric permittivity and magnetic permeability, mo eo ¼ c2 , and some vector identities and the other Maxwell equations have been surreptitiously used. If the material exhibits magnetic or polar characteristics, the situation is slightly more complicated, but such complications are ignored in the present analysis. The relation between current J and electric field E is given by J ¼ sE so that (


 ) 1 @ 2 s @E =  2 : E¼ c @t eo c2 @t 2

ð387Þ

ELECTRON EMISSION PHYSICS

155

Taking E to be given by Eo expfiðK  r  otÞg, where K is the propagation constant and o is the frequency, then o2 so ¼ i 2 : ð388Þ K 2 þ c eo c The complex refractive index ^ n (the caret denoting a complex quantity) is then defined by
 o s 1=2 o K ¼ 1þi  ^n : ð389Þ oeo c c In free space (s ¼ 0), the familiar relation c ¼ o/K follows, but the presence of resistance (inverse conductivity) implies a dampening due to the imaginary part of the complex refractive index ^ n  n þ ik. An electron accelerated by an electric field over a distance L ¼ v dt increases its energy by (qEv)dt. For a density r of electrons, the power absorbed from the electromagnetic wave heating the conductor is given by qrvE ¼ JE ¼ sE 2 for normal incidence and electron motion parallel to the field. It therefore follows that the length scale d characteristic of power absorption is n h io1 @z ln jEðzÞj2 ¼ c=2ko ¼ l=4pk  d; ð390Þ a quantity known as the penetration depth. (A word of notational caution: in contrast to past notation, k is the imaginary part of ^n, and not a momentum term, here.) Regarding how much power actually enters the metal, for simplicity, consider normal incidence (off‐angle incidence is a staple of textbooks and readily found elsewhere). For the electromagnetic wave, the amplitudes of the electric and magnetic components must be equal at the interface. If the electric field is E ¼ ^iEo expfiðKz  otÞg (where ^i is the  unit vector along the x‐axis), then from Maxwell’s equations H ¼ ^j oc K Eo expfiðKz  otÞg. The equations relating the amplitude of E and H become, in matrix notation (recall the quantum tunneling problems for which the present problem bears a passing similarity), 

inc 

Eo 1 1 Eotrans 1 1 ; ð391Þ ¼ K inc K inc 0 K^ trans K^ trans Eorefl where the superscript denotes whether the wave is incident, transmitted, or reflected. Because the incident (or LHS) medium is assumed to be vacuum, K inc is real; similarly, because the RHS is a metal, K^ trans is complex, as is

156

KEVIN L. JENSEN

indicated by the caret. These expressions allow Eorefl to be expressed in terms of Eoinc, and it is found Eorefl ¼

K^ trans  K inc inc Eo : K^ trans þ K inc

ð392Þ

The reflectivity R is the ratio of the absolute square of the magnitudes of the reflected with the incident wave, and so R¼

jK^ trans  K inc j2 j^ n  1j2 ðn  1Þ2 þ k2 ¼ ¼ jK^ trans þ K inc j2 j^ n þ 1j2 ðn  1Þ2 þ k2 :

ð393Þ

Thus, the two parameters that govern how much light is absorbed by a material, the penetration depth d and the reflectivity R, can be ascertained from the complex index of refraction Eq. (393). The question then becomes how the values of n and k are ascertained, which in turn is related to the question of how s is determined.

2. Drude Model: Classical Approach When an electric field is maintained across a material, such as when the gap between a capacitor is filled with a dielectric, the field within the dielectric is less than that which would exist if the gap were a vacuum. If the material is a metal, then electrons would flow to the surface in such numbers as to completely screen out the field within the metal. In dielectrics, the electrons are bound, such that the electron‐ion units deform into dipoles (Figure 59), whose cumulative effect is to partially shield out the external field. The degree to which the electron‐ion unit deforms (i.e., the strength of the dipole) is related to the magnitude of the electric field, and so the polarization P is related to the electric field by P ¼ eowE, where w is the susceptibility and static conditions are assumed. For dielectric materials, therefore, Maxwell’s equations can be retained in form by introducing the D field given by

E=0

E≠0 r

FIGURE 59. Deformation of the ion‐electric cloud by the application of a (vertical) electric field.

ELECTRON EMISSION PHYSICS

157

D ¼ e0 E þ P ¼ e0 ð1 þ wÞE;

ð394Þ

so that =  D ¼ qr, where r is still the number charge density of free (not bound) electrons. In the presence of an electric field that is time varying, the polarization of charge within the dielectric proceeds after changes in the E field occur. Consequently, w acts as a response function and the polarization satisfies ð1 PðtÞ ¼ e0 wðt  t 0 ÞEðt 0 Þdt 0 ; ð395Þ 1

where w(t) is a real function. The Fourier transform of equations of the form of Eq. (395) into frequency space results in P(o) being simply given by the product of the Fourier transforms of each of the integrand functions, or PðoÞ ¼ e0 ^ wðoÞEðoÞ

ð396Þ

(albeit that ^ wðoÞ is not defined with the customary 2p of Fourier transforms so as to retain the form of Eq. (396)) but now, and as indicated by the caret atop w, the susceptibility is no longer necessarily real and will have an imaginary component. The relationship between the susceptibility and the previously considered index of refraction is then shown to be (by the consideration of the wave equation in terms of the polarizability) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ nðoÞ ¼ 1 þ ^ wðoÞ: ð397Þ So, the index of refraction has been expressed in terms of the susceptibility— another quantity that requires a model. Consider first a classical argument for its evaluation, after which a quantum‐based argument is made. In the former, as a result of a time‐dependent electric field, a bound electron oscillates about an atom according to the equation m@t2 rðtÞ þ ðm=tÞ@t rðtÞ þ mo2o rðtÞ ¼ qEðtÞ;

ð398Þ

where the term mo2o corresponds to a restoring force, m=t corresponds to a dissipation or dampening term (moving electrons both radiate and scatter)— and t therefore a relaxation time, and the bound electron is treated as a harmonic oscillator. The Fourier transform of r(t) is rðoÞ ¼

1 qEðoÞ  2 o  o2o þ iðo=tÞ : m

ð399Þ

The induced (atomic) dipole moment is the product of the electron charge with r(o), and it is also equal to the product of the atomic polarizability with the electric field. The macroscopic polarizability P is the sum over all such atomic ones, of which the number density is ro, and so (the sign change being due to the negative electron charge)

158

KEVIN L. JENSEN

wðoÞ ¼ 

1 q2 ro  2 o  o2  iðo=tÞ eo m o o2p

ð400Þ

o2o  o2  iðo=tÞ

where the plasma frequency is defined by  1=2 : op ¼ q2 ro =eo m

ð401Þ

For ro characteristic of the number density of metals, or 1023 atoms/cm3, the plasma frequency is on the order of 1.784
1016 rad/s (UV regime). For a metal, there is no restoring force, meaning the electrons are free to move about so that oo ¼ 0. Consequently, ^ wÞ þ i Imð^wÞ; n2 ¼ 1 þ Reð^

ð402Þ

or, in terms of the real and imaginary parts, n2  k 2 ¼ 1  2nk ¼

o2p t2 ð1 þ o2 t2 Þ

o2p t oð1 þ o2 t2 Þ

Defining n2  k2 ¼ N1 , 2nk ¼ N2 , it can be shown 8 91=2 <1 h i=  1=2 N12 þ N22 n¼ þ N1 :2 ; 8 91=2 <1 h  i=  1=2 N12 þ N22 k¼  N1 :2 ;

ð403Þ

ð404Þ

A representative case loosely based on copperlike parameters is instructive. A conductivity of s ¼ 5.95
105 (O cm)1 plus a number density of ro ¼ 8.411
1022 cm3 (corresponding to m ¼ 7 eV) entails t ¼ 25 fs and op ¼ 1.64
016 1/s. 3. Drude Model: Distribution Function Approach The distribution function approach to the evaluation of number and current density provides another avenue. Consider smoothly varying electric fields that change over length scales that are comparatively long, so that the Wigner function is a solution to the Boltzmann equation. Although there is

ELECTRON EMISSION PHYSICS

159

no need to restrict attention to one dimension, as the 3D case is straightforwardly the same, it is simply a matter of convenience. Thus, recall ð q 1
hk f ðkÞdk; ð405Þ J¼ 2p 1 m where f(k) satisfies _ k f ¼ @c f ; _ x f þ k@ @t f þ x@

ð406Þ

where the dots indicate time derivatives and @ c f is the scattering term. With x_ ¼
hk=m and
hk_ ¼ F (the velocity and acceleration, respectively) and invoking the relaxation time approximation, then @t f þ


k h F 1 @x f  @k f ¼  ð f  f0 Þ; m
h t

ð407Þ

where f0 is the equilibrium distribution in the absence of fields and temperature gradients. Consider turning on the field F gradually using a parameter l (not a wavelength), which changes from 0 to 1, that is, let F ) lF . It follows that the distribution function will likewise ‘‘turn on’’ to the full distribution via a series of progressively smaller terms characterized by the power of l. In other words, f ) f0 þ lf1 þ l2 f2 þ . . . :

ð408Þ

Inserting Eq. (408) into Eq. (407) and equating like powers of l results in for l0 @t f0 þ


k h @x f0 ¼ 0; m

ð409Þ

which is a restatement of equilibrium: the distribution does not vary spatially in the absence of forces or with time and depends at most on k. The time independence of f0 implies, as per the continuity equation @t r þ @x J ¼ 0 that f0 is a symmetric function in k: it is tantamount to the supply function in the derivation of the Richardson and FN equations. The next power is l1, for which @t f1 þ


hk F 1 @ x f1  @ k f0 ¼  f1 : m h
t

ð410Þ

Recall that along the trajectories of the equilibrium distribution, the energy does not change (e.g., the harmonic oscillator treated in the Wigner distribution approach surrounding Eq. (160)). It is therefore reasonable that the effect of the field will be to change the energy, and therefore f1 will be proportional to an energy‐like term. In fact, viewing f1 as the first term in a Taylor expansion implies

160

KEVIN L. JENSEN

f1 ¼ f  f0 ¼ Fðx; k; tÞð@E f0 Þ;

ð411Þ

where F is to be determined (and is not to be confused with work function). Inspection of Eq. (410) suggests that F has the same spatial and temporal dependence as F ¼ Fo expfiðKx  otÞg, but with a k‐dependent coefficient, or Fðx; k; tÞ ¼ Fo ðkÞexpfiðKx  otÞg:

ð412Þ

Coupled with the relation @k f ¼ ð@k EÞð@E f Þ, where E is the energy, Eqs. (411) and (412) allow Eq. (410) to be written 

hk Fo 1 io þ i K Fo ðkÞ  ð@k EÞ ð@E f0 Þ ¼  Fo ðkÞð@E f0 Þ: ð413Þ m t h
With the assumption that the energy is parabolic in k and that, first, K ¼^ no=c and second, that
hk=m  c (i.e., the electron velocity is much smaller than the speed of light), Eq. (413) entails
 tFo hk
Fo ðkÞ ¼ : ð414Þ 1  iot m Finally, using Eq. (405), the definition J ¼ sðF =qÞ (a rather peculiar way of stating Ohm’s law given that F is a force) and the zero‐temperature limit of the supply function entails 0 12 0 1 ð qJ q2 kF tFo @
hkA @ m A ¼ dk sðoÞ ¼ F 2p kF 1  iot m p
h2 ð415Þ tq2 k3F 1 ¼ 2 ð1  iotÞ 3p m Using the relationship between density ro and kF, the expression for direct current (DC) conductivity, and Eq. (389) it follows sðoÞ ¼ sð0Þ

1 þ iot ; 1 þ o2 t2

ð416Þ

from which Eq. (403) follows. For metals, the scattering time t is on the order of 10 fs; therefore, for optical frequencies (e.g., a wavelength of 532 nm), ot is on the order of 35. The distribution function approach points out that the Drude relations break down when higher‐order l terms are nonnegligible (large fields), the relation between k and E is more complex than assumed, or temperature gradients add complications. If these complications can be ignored (for the present, they can), the consequences of Eq. (403) are twofold. First, in the limit of vanishing frequency, n and k become approximately equal, and the reflectivity R approaches

ELECTRON EMISSION PHYSICS

sffiffiffiffiffiffiffiffiffiffi 2e0 o lim R ¼ 1  2 ; ot!0 sð0Þ

161 ð417Þ

which is known as the Hagen–Rubens equation; that is, at low frequencies (wavelengths longer than infrared), the metal is nearly 100% reflective (R ¼ 1). In the opposite limit, n approaches unity and k approaches 0, so that R ¼ 0, indicating that metals are transparent in the UV limit (o must be larger than the plasma frequency—for copper, the plasma frequency corresponds to a wavelength of 100 nm, or smaller than the lower limit of the visible spectrum at 400 nm). In the o ! 0 limit, the static dielectric constant results and is defined by ( eð0Þ ¼ e0




op 1þ oo

2 )

:

ð418Þ

In the case of metals, where oo [encountered in Eq. (398)] is vanishingly small, Eq. (418) suggests inordinately large static dielectric constants. For semiconductors, where the physics is a bit different, the same equation suggests more reasonable values. By way of example, consider a semiconductor‐like material (for which number of electrons per atom, effective mass variation depending on crystal orientation, and similar complications are suppressed) nominally modeled after silicon. For a number density (the ratio of the density with the atomic mass) of 5
1022 atoms/cm3, each of which contributes one electron, op is 12.6
1016 rad/s. If oo corresponds to an optical wavelength, such as 500 nm, then oo ¼ 3.77
1015 rad/s, implying that e0 ¼ 12.2, typical of semiconductors—like silicon, not surprisingly. The energy hoo ¼ 2.48 eV is similar in energy to where sharp changes in the absorption
coefficient k occur (Philipp and Ehrenreich, 1963) in the dielectric constant of silicon that are indicative of a contribution such as Eq. (400). The extension of the Drude model to semiconductors is possible, but complications arise from the lower free‐carrier density of electrons compared to metals and the presence of a band gap (Jensen, 1985; Jensen and Jensen, 1991). Importantly, the behavior of the dielectric components is related to the validity of the quasiclassical Boltzmann transport equation widely used to model solids. In the quantum mechanical theory (using either the density matrix approach, or second order perturbation theory approach which gives the same result), the relaxation time is constant at low frequencies but is frequency dependent at high frequencies so that absorption coefficient varies not as the second power of the wavelength as in the Drude model, but as the third or fourth (depending on the scattering mechanisms), though a treatment revealing the dependence is outside the scope of this monograph.

162

KEVIN L. JENSEN

Reflectivity is complicated by the presence of resonances in the visible regime of the spectrum, as such resonances occur for optical frequencies, which is precisely where present interests lie. The drive lasers for photocathodes use harmonics of the fundamental frequency of an Nd:YAG laser (wavelengths of (1064 nm)/n so that n ¼ 3, 4, and 5 correspond to 355 nm, 266 nm, and 213 nm, respectively) or titanium‐sapphire (Ti:Saph) laser (with a fundamental of 800 nm so that n ¼ 2 corresponds to 400 nm). Thus, the reflectivity in the visible regime is of interest. 4. Quantum Extension and Resonance Frequencies Let the charge
electric field in the vicinity of the atom be given by Fo cos(ot) and the ground state of the electron in the absence of the field be desig^ ðtÞ  ^r (where the caret indicates nated fo. As a result of the perturbation F operator), the wave function becomes X   cj ðtÞjfj iexp ioj t ; ð419Þ jcðtÞi ¼ jfo iexpðioo tÞ þ where
hoj is the energy difference between the j th level and the ground state th (o ) level. The coefficients satisfy the relation (a consequence of the time‐ dependent Schro¨dinger equation and the orthogonality of the basis states)

  ^ ðtÞ  ^rjfo iexp i oj  oo t ; ð420Þ i
h@t cj ðtÞ ¼ hfj jF the solution is (for sake of argument the electric field is taken to be along the x‐axis) ð

 1 t cj ðtÞ ¼ Fx hfj j^ xjfo icosðotÞexp iðoj  oo Þt : ð421Þ 2i
h 0 Consequently, the expectation value of the dipole term is n 1  1 o 2qFx X hcðtÞjq^ xðtÞjcðtÞi ¼ jhfj j^ xjfo ij2 oj  o þ oj þ o cosðotÞ: h
j ð422Þ where the terms that oscillate rapidly and out of phase with the field have been ignored. The dipole moment of the atom is therefore proportional was assumed previously). Use the relation to the electric  field (which 1 1 2 2 oj  o þ oj þ o ¼ 2o j =ðoj  o Þ and introduce the oscillator  2 2 strengthPterm fj ¼ 2m=
h
xjfo ij : The atomic polarizability is then hoj jhfj j^ ðq2 =mÞ j fj =ðo2j  o2 Þ: Summing over all such terms in a unit area then shows that the quantum extension of Eq. (340) is (where the dampening term has been reinserted by hand)

163

ELECTRON EMISSION PHYSICS

wðoÞ ¼

fj q2 ro X  : j eo m 2 o  o2  iðo=tÞ

ð423Þ

j

Some subtleties have been glossed over between the local and the macroscopic fields, for which the reader is referred to Ziman (1985, 2001). How much of a consequence does Eq. (423) make? Consider a simple pedagogical example based on copperlike parameters, for which
hop ¼ 10:769eV. If the conductivity is taken to be s ¼ 5.959
105 (O‐cm)1, then t ¼ 25.144 fs. Values of n and k (for example, from the CRC tables; Weast, 1988) at normal incidence, from which R can be obtained, are shown in Figure 60 on the line designated ‘‘Exp.’’ Similarly, the behavior of n and k as calculated using Eqs. (403) and (404) is shown by the line designated ‘‘Drude.’’ Clearly, while the Drude line captures the changeover from reflective to transparent, some physics is missing. An ad hoc Lorentz term using the parameters f1 ¼ 1,
ho1 ¼ 5:9239eV and t1 ¼ 0.25 fs (being a resonance line, the value of t1 will not necessarily equal t) is shown in the line f1, from which it is seen much of the difference is captured. Likewise, choosing the somewhat arbitrary parameters f1 ¼ 4/5, f2 ¼ 1/5, o2 ¼ 2o1, and t2 ¼ t1 improves the correspondence by addressing the tail. Clearly, an optimized fitting procedure using Lorentzian components will have good success capturing the behavior of R(o). Such an exercise is of pedagogical, but for purposes herein not practical, interest—in assessing the role of reflectance in diminishing QE, it is simpler to extract the values of R from actual optical constant data available from the literature. In the event that 1.0

Reflectance

0.8 0.6 0.4 0.2 0.0

Exp. Drude f1 f1 & f2 1

10 Energy [eV]

FIGURE 60. Comparison of experimental reflectance to the Drude model and two Lorentzian terms.

164

KEVIN L. JENSEN

the incident light is not normal to the surface, then the off‐angle formulas allow the reflectance to be obtained (Gray, 1972). The relations Rs ¼

a2 þ b2  2a cosy þ cos2 y a2 þ b2 þ 2a cosy þ cos2 y

Rp ¼ Rs

a2 þ b2  2a siny tany þ sin2 y tan2 y

a2 þ b2 þ 2a siny tany þ sin2 y tan2 y  1 R ¼ Rs þ Rp 2 d¼

ð424Þ

l 4pk

where y is the angle from normal, and the terms a and b are given by h i1=2 2a2 ¼ ðn2  k2  sin2 yÞ2 þ ð2nkÞ2 þ ðn2  k2  sin2 yÞ h i1=2 2b2 ¼ ðn2  k2  sin2 yÞ2 þ ð2nkÞ2  ðn2  k2  sin2 yÞ

ð425Þ

allow the value of R for several common metals to be obtained (Figure 61). It is observed that the maximum absorption can occur off normal, as with tungsten and lead.

Reflectivity [%]

100

80

60

40

0

30

60

90

Angle [deg] Cu Ag

Au Pb

W

FIGURE 61. The reflectivity of various metals as a function of incidence angle using Eq. (424).

ELECTRON EMISSION PHYSICS

165

E. Conductivity The scattering rate factor appearing in the Drude model and alluded to in the discussion of the mean free path is referred to as the relaxation time in the discussion of the Wigner and Boltzmann equations. Electrons moving through a solid may interact among themselves or scatter off of phonons and thermalize with the lattice. Such a bland observation in fact entails a great deal of physics relevant to photoemission models. The probability that a photoexcited electron can transport to the surface depends on its scattering possibility and the temperature of the electron gas, as scattering rates are temperature dependent, which in turn depends on the amount of laser energy absorbed by the material. Photocathodes of the variety considered herein must produce a great deal of current on demand, unlike their photodetector brethren. In fact, the peak current densities demanded of photocathodes for particle accelerators is enormous (on the order of kiloamps, albeit over very short times) and if the QE is poor, then a considerable incident laser intensity and therefore significant heating occur. Scattering affects how the laser energy is distributed into the lattice, which in turn affects the electron temperature, which in turn affects the scattering rate. The scattering factors are therefore not simply parameters to be inferred from, say, electrical conductivity, but merit consideration in their own right. The evolution of the distribution model represents a convenient starting point. In keeping with the approach so far, the concern is when electron flow is in the direction of applied fields (if any), and insofar as a surface exists, it is normal to the direction of electron transport. This bland assumption indicates that the problem at hand can be treated as a 1D problem, but what is stated here can easily be expressed in full 3D parlance. The only motive for one dimension is purely the argument of ease, but this rationale has much to commend it. Many excellent sources consider the problem in its full 3D glory but arrive at the same conclusions (Hummel, 1992; Ibach and Lu¨th, 1996; Kittel, 1962), albeit via a more rigorous and therefore arduous route. 1. Electrical Conductivity Reconsider the linearized Boltzmann equation for a time‐independent equilibrium distribution, for which the scattering term vanishes (as many electrons scatter into a state as scatter out). The equilibrium distribution would then nominally appear to satisfy
k h F @x f ðx; kÞ þ @k f ðx; kÞ ¼ 0: m h


ð426Þ

166

KEVIN L. JENSEN

A naive solution to Eq. (426) would seem to be 

m f ðx; kÞ  fS ðx; kÞ  ln 1 þ exp b ð mðxÞ  EðkÞ Þ ; pb
h2

ð427Þ

where mðxÞ  m þ jðxÞ, F ¼ @x j; and the subscript on f reinforces that the result is symmetric in k. But this cannot be correct; only the antisymmetric component ( fA) of a distribution in k gives rise to an overall current. Consider, however, how many electrons are actually required to produce an appreciable current. In the Richardson equation—the number of electrons per unit time and area passing over the barrier compared to the number hitting the barrier is phenomenally small: ð1
 hk
! fo ðkÞdk m 4p2
h3 JRLD ko ð1
 : ð428Þ  JRLD  hk
qmm2 Jmax fo ðkÞdk m 0 For m ¼ 7 eV and even for JRLD ¼ 100 A/cm2, the ratio in Eq. (428) is approximately 2.52
1010 because Jmax ¼ 3.96
1011 A/cm2 for copper. A similar analysis, using the FN equation, yields a larger ratio due to the higher current densities, but the conclusion essentially remains the same: the ratio is negligible. The antisymmetric part is missing in Eq. (427), which prevents it from being used to ascertain currents. Nevertheless, the naive approach lends credence to the concept of a ‘‘local’’ chemical potential m(x), elsewhere termed the electrochemical potential, which may be used as long as the potential variation is weak in some sense. In the following discussion, this can be implicitly assumed to have been done. From the definition of current density and Eq. (407) then ð q 1
hk JF ¼ ð f ðx; kÞ  fo ðx; kÞÞdk 2p 1 m 0 1 ð429Þ ð qt 1
hk @
hk F A @x f1 þ @k f0 dk ¼ 2p 1 m m
h where the F subscript (F) reinforces that it is current due to electric field, the first line is a consequence of the symmetry of fo, and the approximation that the relaxation time is independent of k (while not strictly true) is used. In the second line, and recalling Eq. (410), the second term in the brackets is dominant. Neglecting the first term and integrating by parts yields ð qtF 1 qtF qtF 3 r  J¼ k ; f0 ðkÞdk ¼ ð430Þ 2pm 1 m 0 3p2 m F

ELECTRON EMISSION PHYSICS

167

where the zero‐temperature approximation to the bulk number density has been used. This identifies the DC conductivity s ¼ qJ/F as sð0Þ ¼

q2 t 3 k ; 3p2 m F

ð431Þ

where the scattering rate is now identified as the relaxation time. The approach leading to Eq. (431) is now recapitulated for gradients in temperature to reveal a fundamental relation between electrical and thermal conductivity. 2. Thermal Conductivity The energy carried by the electrons during their migration is transferred to the lattice when they scatter. The question arises: What is the thermal conductivity of an electron gas, that is, the proportionality between current and temperature (as opposed to potential) gradient? A related concept is specific heat, or the change in energy with temperature. Consider by way of introduction the statistical mechanics of an ideal gas. In a cubic box, a third of the particles are moving along each of the coordinates, and half of those are in the plus (þ) direction with the other half in the minus (–) direction. If the particles share the same velocity v, then the number per unit area that impact the face of the cube in a time t is rvt=6, where r is the number density. After impact, the change in momentum is 2 mv. Therefore, the pressure P is the product of density and momentum transfer per unit time, and so P ¼ rmv2 =3. The quantity mv2 =2 is the kinetic energy. The ideal gas equation plus the expression for kinetic energy combine to show that the energy of a particle in terms of temperature is 32 kB T, and it follows that the energy density of the ideal gas of particles is 3 E ¼ rkB T: 2

ð432Þ

The coefficient of T in Eq. (432) is the specific heat capacity Cv ¼ ð3=2ÞrkB : If there is a temperature gradient, in the short distance v dt that the particles travel between thermalizing collisions, the flow of energy is the difference in the number per unit area of the particles from the left with their unit energy, and the particles from the right 0 1 vt dE vt @dxA 3 dTðxÞ 1 2 dT JT ¼ ¼ rkB ¼ rv tkB 6 dt 6 dt 2 dx 2 dx ð433Þ 1 dT 2 dT K ¼ Cv v t 3 dx dx where vt is the length between collisions and the (1/6) comes from the arguments above relating to how many particles pass through a given face.

168

KEVIN L. JENSEN

An alternate method of writing the relationship, assuming that only the electrons at the Fermi level matter in heat flow—and this remains to be shown—is given by K ¼ ð2m=3mÞCv t. Regrettably for the derivation just presented, electrons do not have a uniform energy and therefore do not travel at a uniform velocity. However, Eq. (433) provides an indication of what must occur, and so the problem is reconsidered afresh from the distribution function approach now that there is some confidence in the destination. (Remember that the transport of heat is the issue.) When it was charge that was transported, the current density was the product of the charge, its velocity, and the density of charge. Now, however, not charge but energy is flowing. From thermodynamics a quantity of heat dQ obeys dQ ¼ dU  mdr:

ð434Þ

In a zero‐temperature equilibrium distribution, any small quantity of energy added to a particle is added at the Fermi level. The flow of heat is therefore the net imbalance of a particle traveling in one direction with an energy E compared to its matching particle traveling in the opposite direction at the Fermi level. In equilibrium, there is no net flow (dQ ¼ 0), and the disturbance from equilibrium implied by Eq. (434) is small and affects but one particle. Using the relationship dU ¼ EðkÞfFD ðEðkÞÞdk then dJ E ¼ EðkÞð
hkx =mÞfFD ðEðkÞÞdk, where, out of necessity, the full 3D approach reappears and the shorthand dk ¼ dkx dky dkz ¼ 4pk2 dk is useful. The number current density is, as before, dJ e ¼ ð
hkx =mÞ fFD ðEðkÞÞdk: Consequently, the current of heat JQ is then related to the current of energy by JQ ¼ JE  mJe 0 1 ð   hkx
¼ ð2pÞ3 EðkÞ  m @ AfFD ðEðkÞÞdk m 0 12 ð hkx
3 ðEðkÞ  mÞtðEðkÞÞ@ A ½@x fFD ðEðkÞÞdk ¼ ð2pÞ m

ð435Þ

In the last line, the term containing the equilibrium distribution vanishes by the asymmetry of the integrand in kx. Recalling the density of states D(E) defined by Eq. (30), Eq. (435) may be written in a more general form that allows for nonspherical Fermi surfaces in terms of which 0 12  ð1 hkx
ðE  mÞ@ A tðEÞDðEÞ @x fFD ðEÞ dE JQ ¼ m 0 8 9 ð436Þ < 2 ð1 = ¼ ðE  mÞEtðEÞDðEÞ½@T fFD ðEÞdE ð@x TÞ :3m 0 ;

169

ELECTRON EMISSION PHYSICS

where Ex ¼ E/3 by spherical symmetry. The coefficient of @ xT is the thermal conductivity. To proceed further [i.e., reclaim Eq. (433)], the nature of @T fFD must be investigated. The gradient with respect to temperature of the FD distribution, for any reasonable temperature encountered in practice, is a sharply peaked function. Letting u ¼ bðE  mÞ in the FD distribution, then @T fFD ðEÞ ¼ kB b

u ð1 þ

eu Þð1

þ eu Þ

:

ð437Þ

For a smoothly varying function g(du), a Taylor expansion and Appendices 1 and 2 can be used to show that for small d ð1 gðduÞ 2d2 00 2d4 0000 du  gð0Þ þ g ð0ÞWþ ð2; 1Þ þ g ð0ÞWþ ð4; 1Þ u u 2! 4! 1 ð1 þ e Þð1 þ e Þ 7 ð438Þ ¼ gð0Þ þ d2 g00 ð0Þzð2Þ þ d4 g0000 ð0Þzð4Þ 4 where primes indicate derivatives with respect to argument, the function W (n,x) is discussed at length in Appendix A, and z is the Riemann zeta function discussed in Appendix B. The leading‐order term justifies the standard approximation that @T fFD ðEÞ mimics a Dirac delta function when mixed with slowly varying functions over the range jE  mj kB T. Any smoothly varying function in E can have its argument replaced by m and pulled from the integral, letting Eq. (436) be approximated by ð 1  2m JQ ¼ tðmÞ@T ðE  mÞfFD ðE ÞDðE ÞdE ð@x T Þ 3m 0 ð439Þ 2m tðmÞCe ðT Þð@x T Þ  kð@x T Þ  3m Eq. (439) is formally equivalent to Eq. (433) if v2 ¼ 2m=m and t ¼ tðmÞ, that is, the velocity and relaxation time are evaluated at the Fermi level. The specific heat can be further approximated by ð 1  @ Ce ðTÞ  ðE  mÞ fFD ðE ÞDðE ÞdE @T 0 8 0 1 0 19 < = u u u2 D@m þ A þ D@m  A ð bm : b b ; ð440Þ 2  kB T du u u ð1 þ e Þð1 þ e Þ 0 8 9 < = D ð m Þ  k2B T 2DðmÞWþ ð2; bmÞ  W ð 4; bm Þ þ : ; ð2bmÞ2

170

KEVIN L. JENSEN

where Eq. (A4) is invoked and bm  1 has been used. For temperatures of present concern, only the leading‐order term in Wþ(2,1) ¼ z(2) matters, and so 8 0 12 9 < = 1 7 p Ce ðT Þ ¼ p2 k2B TDðmÞ 1  @ A : 3 40 bm ; ð441Þ p2 2  kB DðmÞT  gT 3 where k2F ¼ 2mm=
h2 . The theoretical value of g differs, in general, from the experimental value; given the linear dependence of g on m, it is common to define a ‘‘thermal’’ mass by


gexp mth ¼ : m gtheory

ð442Þ

Several examples are
(Cu) ¼ 1.375,
(Ag) ¼ 1.0136, and
(Al) ¼ 1.4855, indicating that the simple model works reasonably well. The difference between the thermal mass and the electron mass is attributed to the influence of the periodic potential of the atoms on electron motion, as well as their vibration modes (phonons), which is discussed in more detail below. For other metals, it is found that
(Fe) ¼ 7.931, and other transition metals are comparably high. In these cases, the partially filled d shells contribute to the density of states (DOS) and thereby undercut the simple model put forward here (Ibach and Lu¨th, 1996). 3. Wiedemann–Franz Law Let us now compare the electrical conductivity to the thermal conductivity under the assumption that the thermal and electrical relaxation times are the same—an assumption that is not a priori obvious. We find !
 2
h2 k2F m 2 k k T tðmÞ
 B F 3
h2 k 3m 2m p2 kB 2 ¼ ¼ T  LT; ð443Þ q2 t ð m Þ 3 s 3 q k 3p2 m F where L ¼ 2.44301
108 ohm‐watt/Kelvin2 is the Lorentz number. The empirical Lorentz number Lexp  kexp =sexp for various metals compared to the theoretical value L is shown in Figure 62. Implicit in Eq. (443) is the assumption that the relaxation time for thermal processes is equivalent to that for electrical conduction. Such a circumstance is not a priori true, as electrical currents resemble a displacement

ELECTRON EMISSION PHYSICS

171

3.0

L [10−8 W-Ω/K2]

2.8

Experiment Theoretical

2.6 2.4 2.2 2.0

Cu Au Ag

Al W Mo Na Pb Element

FIGURE 62. Theoretical (line) versus experimental Lorentz numbers for various metals.

of the Fermi sphere, but thermal effects affect the distribution in electron energy near the Fermi level. The point, however, is that even though L for a variety of metals is not the Lorentz number, it is remarkably close. A final comment concerning the electrical and thermal conductivities is that, as generally observed, good thermal conductors make good electrical conductors because energy is transported by free electrons. For insulators, where free electrons are scarce, the absence of electrical current is related to the fact that heat is transported phonons, so that poor conductors are related to thermal insulators as well. 4. Specific Heat of Solids Thermal energy is also related to motion of the lattice; atoms in the lattice vibrate and the degree of their vibration relates to the temperature of the solid. Vibrations constitute harmonic oscillators, for which the average kinetic energy is the same as the average potential energy. A simple model is that the total energy held by the harmonic oscillators representing the lattice is E ¼ 2ð3=2ÞkB T ¼ 3kB T. For N atoms per mole, then, the total internal energy is 3NkB T. The coefficient of temperature, identified as heat capacity per unit volume, is then Ci ¼ 3ðN=V ÞkB ¼ 3ri kB for solids—an i subscript (i) is used to distinguish the contribution to specific heat from phonons from that due to electrons. At low temperatures the relation fails, and a quantum mechanical treatment must be considered. Treat the atoms in the lattice as a set of coupled quantized harmonic oscillators. If they are in contact with a heat bath of temperature T,

172

KEVIN L. JENSEN

then the probability that the harmonic oscillator will be at a level En ¼ ðn þ 1=2Þ
ho 
hon is proportional to expð
hon =kB T Þ ¼ expðb
hon Þ. The constant of proportionality is obtained by observing that the sum over all probabilities must be 1, or X1 P b
ho P1 nb
ho ho n ¼ N eb
P n ¼ No 1 o 0 e 0 e 0 b
h o=2 No e ð444Þ ¼ ¼1 1  eb
ho which defines No. The probability that the oscillator at the energy level n is   ð445Þ Pn ¼ enb
ho eb
ho  1 : The mean energy of the system is the sum over the products of the energy levels En with their probability Pn, or
 X1  b
ho  X1 1 nb
ho E P ¼ e 1
ho nþ e : ð446Þ E ðoÞ ¼ n¼0 n n n¼0 2 The sums are readily evaluated by X1 X1 n 1 ¼ x  xn n¼0 n¼1 X1 X1 X1 n n ¼ x  x x ¼ ð 1  x Þ xn n¼0 n¼0 n¼0 and tricks such as

X1

nxn ¼ @x n¼1

X1 n¼0

xn ¼

x ð1  xÞ2

:

ð447Þ

ð448Þ

Both terms on the RHS of Eq. (446) may then be evaluated, giving
 1 1 : ð449Þ E ðoÞ ¼
ho þ b
ho 2 e 1 The recasting of E(o) as ðhni þ 1=2Þ
ho identifies hni ¼

1 ; 1

eb
ho

ð450Þ

that is, the oscillators obey BE statistics. A complication is the realization that a complex arrangement of atoms in a crystal does not oscillate with only one fundamental frequency o; rather, N unit cells with r atoms per unit cell oscillating along three axes implies 3r N modes, all of which can be excited. For an isotropic crystal of volume V, the number of modes per unit volume in frequency space is constant. Given that the number of atoms present in even a paltry bit of matter is enormous,

ELECTRON EMISSION PHYSICS

173

the summations can be converted integrals without difficulty. The DOS Dp(o) (p designating phonon) in reciprocal space q will then be
 1 1 1 o2 3 2 Dp ðoÞdo ¼ d q¼ 4pq dq ¼ 2 do; ð451Þ 2p c3i ð2pÞ3 ð2pÞ3 where o=q ¼ ci is the sound velocity for the ith branch. If the transverse T sound velocities are assumed to be the same (but different than the longitudinal L), then
 V 1 2 Dp ðoÞdo ¼ 2 3 þ 3 o2 do: ð452Þ 2p cL cT The total energy of the crystal is then a summation (now integration) over all frequencies of the energies and the DOS, or ð oD ð453Þ E ¼ Dp ðoÞEðoÞdo; 0

where oD is the maximum, or Debye cutoff, frequency determined from the requirement that the number of modes is equal to the number of atoms (concepts discussed in greater detail in the treatment of the electron‐lattice relaxation time), or
 ð oD V 1 2 3rN ¼ Dp ðoÞdo ¼ 2 3 þ 3 o3D : ð454Þ 6p cT cL 0 It is more common to speak of a Debye temperature TD defined by the relation
hoD ¼ kB TD and estimated from TD ¼


hvs  2 1=3 6p Nr ; kB

ð455Þ

where be ¼ 1/kBTe is the electron temperature thermal factor, N [#/cm3] is the number density of the crystal, r is the number of atoms per unit cell, and vs is the velocity of sound. Values among metals vary; examples are 165 K for gold, 343 K for copper, and up to 400 K for tungsten. Introducing the lattice density ri ¼ N/V and combining the components (note that o=2, being temperature independent, does not contribute) gives ð 9rri d oD 2
ho do Ci ðTÞ ¼ 3 o b
ho e 1 oD dT 0 0 13 ð yD =T T x4 ¼ 9rkB ri @ A dx ð456Þ x x yD 0 ð1  e Þð1  e Þ 0 1 0 13 T y D ¼ 9rkB ri @ A W @4; A yD T

174

KEVIN L. JENSEN

Ci(T)/Ci(∞)

1.00

0.1 Exact Small approx Large approx 0.01 −3

−2

−1

0 In(T/q D)

1

2

3

FIGURE 63. Comparison of Ci (T) calculated numerically with its asymptotic limits.

For T  yD , then W ð4; xÞ  x3 =3: the classical result is regained (for simple metals, r ¼ 1). However, for T  yD then W ð4; xÞ  4p4 =15, and Ci ðT  yD Þ  3rkB ri ð4p4 =5ÞðT=yD Þ3 . The ratio of Ci(T) with the classical 3 rrikB is shown in Figure 63. The total specific heat for a solid is then the sum of the electron and lattice components. F. Scattering Rates The resistivity r of a metal is known to be temperature dependent, a common relation being the linear form of rðTÞ ¼ ro þ aðT  To Þ;

ð457Þ

where T is temperature, To is a reference temperature (commonly room temperature), and a is the temperature coefficient. As resistivity is the inverse of conductivity, Eq. (457) implies that the relaxation time is temperature dependent. To determine the dependence, evaluating the relaxation times must be done using quantum transport theory (Jensen et al., 2007; Rammer, 2004; Ridley, 1999; Wagner and Bowers, 1978). The strategy here shall be to provide descriptions of sufficient plausibility that when complex relations are provided (deus ex machina), they are plausible. 1. Fermi’s Golden Rule ^ where Let the Hamiltonian of a scattering problem be given by H^ ¼ H^0 þ U, ^ ^ U represents the scattering potential and H0 the unperturbed Hamiltonian. Define the unperturbed wave functions by

175

ELECTRON EMISSION PHYSICS

P

jcðtÞi ¼ k ck ðtÞjck ðtÞi ¼ H^0 jki ¼ E ðkÞjki

P

k ck ðtÞjkie

iok t

ð458Þ

Consider the scattering off of a weak potential in that scattering to other k states is not large. Then it follows that Schro¨dinger’s equation   i
h@t jcðtÞi ¼ H^0 þ U^ jcðtÞi ð459Þ that i
h @ t c k0 ¼

X

^ c hk0 jUðtÞjkiexp f i ð o k0 k k

 ok Þtg;

ð460Þ

where k0 denotes the state after scattering from the state k, which is the object of our attention. Although the potential may have a complex dependence on time, it is easier to consider its Fourier components, and so assume ^ ¼ U^ e iot : the sign on o will (shortly) indicate if a particle is emitted UðtÞ o or absorbed (when discussing phonons). If the initial state k0 is empty, then an integration of Eq. (460) gives  iX expfiðok0  ok oÞtg  1 0 ^ ck0 ðtÞ ¼  c hk jUo jki : ð461Þ k k h
ðok0  ok oÞ In the Born approximation, one initial state ko dominates all others; therefore all but one of the ck vanish—that is, ck  dk;ko . In deference to keeping terminology manageable, the ‘‘o’’ subscript will be ignored (i.e., ko ¼ k). It follows  expfiðok0  ok oÞtg  1 0 ^ ck0 ðtÞ  hk jUo jki : ð462Þ i
h ð o k0  o k o Þ The probability Pk(t) that a scattered electron ends up in the state k is thenjck ðtÞj2 , for which Eq. (461) is less than computationally elegant. As when evaluating density, current, or other moments of a distribution function, an integration over the final states is performed, entailing an examination of how Eq. (461) behaves. Let O ¼ ðok0  ok oÞ and eiOt  1 sinðOt=2Þ ¼ 2eiOt=2 : iOt Ot

ð463Þ

For large times t, the absolute square of the RHS of Eq. (462) is sharply peaked,Ð indicating that integrations with it are concentrated about O ¼ 0. 1 Using 1 x2 sin2 ðxÞdx ¼ p, then sin2 ðOtÞ ðOtÞ

2



2p dðOÞ; t

where d(x) is the Dirac delta function, and so

ð464Þ

176

KEVIN L. JENSEN

2

0

jck0 ðtÞj  jhk

jU^o jkij2




 2pt dðEðk0 Þ  EðkÞ
hoÞ: h


ð465Þ

The notation
ho in the far RHS represents either the absorption or emission of something, and so the sum of each case should be considered in Eq. (465). The probability that a transition has taken place at time t is less interesting in the present circumstance than the rate at which transitions take place (that is, scattering), and it is common to define the latter as the large time limit of jck0 ðtÞj2 =t  S ðk; k0 Þ; using Eq. (465) this yields 0 1 2p n Sðk; k0 Þ ¼ @ A jhk0 jU^oþ jkij2 dðEðk0 Þ  EðkÞ þ
hoÞþ h
ð466Þ o jhk0 jU^o jkij2 dðEðk0 Þ  EðkÞ 
hoÞ As noted by Ridley (Chapter 3, 1999), the delta function is an approximation that applies provided the time between collisions is much longer than the duration of a collision. The relaxation time is then X 1 ¼ S ðk; k0 Þ: k0 tðkÞ

ð467Þ

Equation (467) works well for nondegenerate semiconductors where the concentration of carriers is not great. However, for metals and degenerate semiconductors, the occupation of the final state factors in, as does the occupancy of the initial state. These complications are considered explicitly when treating metals. While Eq. (467) is an estimation of how often collisions occur, what is additionally important is the rate at which both momentum and energy are randomized through collisions. If scattering is not isotropic, the memory of the initial direction (say in the z‐direction) of an initial electron beam takes longer to dissipate, and an additional multiplicative factor equal to the fractional change in the forward, or ^z, momentum is required, or ^z  ð
ðk  k0 Þ k0 hk 
h k0 Þ ¼1 cosy; ¼ 1  ^z 
hk k2 k

ð468Þ

where k ¼ k^z. If scattering is elastic and energy parabolic in momentum, then (k0 /k) is unity. Finally, the fractional change in energy is
0 2 Eðk0 Þ k ¼1 : 1 EðkÞ k

ð469Þ

ELECTRON EMISSION PHYSICS

177

The momentum tm and energy tE relaxation times (Lundstrom, 2000; Ridley, 1999) are defined through the inclusion of the factors given by Eqs. (468) and (469) in Eq. (467), respectively.

2. Charged Impurity Relaxation Time A time‐independent shielded Coulomb potential of the form encountered in Eq. (32) gives rise to hrjU^o jr0 i ¼

Zq2 expðkTF rÞdðr  r0 Þ; 4peo r

ð470Þ

where Zq is the charge of the potential. For degenerate statistics, kTF ¼ ð4kF =pao Þ1=2 [the nondegenerate statistics case is given in Eq. (33)]. Elementary evaluation shows that as kTF becomes large, expðkTF rÞ=r acts like a delta function itself for a smoothly and weakly varying function g(r); that is. to leading order ð1 ekTF r gðrÞdr ¼ 4p gðrÞekTF r rdr r 0 O 0 1 ð 4p 1 @ x A x 4p e xdx  2 gð0Þ g ¼ 2 kTF 0 kTF kTF

ð

ð471Þ

Therefore, before considering the shielded potential, consider first the Dirac delta function approximation, as it illustrates several features of relaxation times without undue burden. It follows 0 1 ð 2 Zq 1 @ A hk0 jridðrÞhrjkidr hk0 jU^o jki ¼ 4pKs eo V k2TF ð1 ð Zq2 2p 1 ¼ dx drdðrÞexpfiprxg 4pKs eo V k2TF 1 0 ¼

ð472Þ

Zq2 Ks eo k2TF V

where x ¼ cosy, V is the volume, and p ¼ jk  k0 j. It is only slightly more difficult to show that (recall Eq. (63), the difference here being the mass of the ionized scattering site)

178

KEVIN L. JENSEN

ð Zq2 expðkTF rÞ hrjkidr hk0 jri r 4pKs eo V ð1 Zq2 ¼ ekTF r sinðprÞdr 4pKs eo V 0 1 Zq2  2 k þ p2 ¼ Ks eo V TF

hk0 jU^o jki ¼

ð473Þ

Eq. (473) reduces to Eq. (472) for kTF » p. Conservation of momentum and energy indicates that the collision is elastic, for which the magnitude of the momentum in the initial and final states are the same; the expression for p is obtained from (where y is now taken as the angle between k and k0 ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ jk  k0 j ¼ k2 þ k02  2kk0 cosy ¼ 2k sinðy=2Þ

ð474Þ

and so hk0 jU^o jki ¼


1 Zq2 y k2TF þ 4k2 sin2 : 2 Ks eo V

ð475Þ

The scattering rate is then P 1 ¼ k0 S ðk; k0 Þ tðkÞ ð 2V 2p 0 ^ jhk jUo jkij2 dðEðkÞ  Eðk0 ÞÞdk0 ) ð476Þ ð2pÞ3
h 0 12 ð ðp 2 @ Zq2 A 1 1 02 0 sinydy 0 ¼ k dk dðEðkÞ  Eðk ÞÞ  2 2 p
h Ks eo V 0 0 kTF þ 4k2 sinðy=2Þ The delta‐function integration is readily dispatched; introducing BðkÞ2 ¼ k2TF =4k2 gives

2 ð p 1 1 2 1=2 Zq2 sinydy ¼ n o2 ; tðEÞ 32p mE 3 K s eo 2 0 BðkÞ þ sinðy=2Þ

ð477Þ

ELECTRON EMISSION PHYSICS

whereas the momentum relaxation time, as per Eq. (468), is

2 ð p 1 1 2 1=2 Zq2 sinyð1  cosyÞdy ¼ n o2 : 3 tm ðkÞ 32p mE K s eo 0 BðkÞ2 þ sinðy=2Þ

179

ð478Þ

The integration can be performed—but note that so far, only one scattering center has been considered, whereas the number of scattering sites (impurity centers) per unit volume ri must be included. Performing the integration and including the scattering sites finally yields

2    1 r 2 1=2 Zq2 B2 2  ¼ i ln 1 þ B : ð479Þ tm ðEÞ 32p mE 3 Ks eo 1 þ B2 The term in brackets is large for semiconductors, of order O(10) to O(100) depending on the effective mass and the ionized impurity concentration (generically about 1017 #/cm3). The coefficient provides a measure of the size of the relaxation time in general: 
1=2
2 2
2
32p mE 3 Zq Ks E½eV 3=2 ¼ 5:1767ps ; ð480Þ ri r 2 K s eo Z where r ¼ m/mo is the effective mass ratio. Therefore, picosecond‐scale relaxation times are to be expected. On occasion it is written ð1=tm Þ ¼ vg sm =V , where sm is a cross section; the inverse scattering rate can therefore be thought of as the ratio of the volume swept out by an area traveling at a group velocity for a scattering time with the cell volume V. The sm calculated via Eq. (480) is often denoted as the Brooks—Herring (BH) approximation—it avoids problems in infinities that arise in the evaluation of a Coulomb potential cross section by appealing to the notion of screening. The BH approach is in contrast to removing the offending infinities by truncating relevant integrals at a lower scattering angle (the Conwell–Weisskopf approximation; Ridley, 1977, 1999). The description of scattering given here relies on the Born approximation, known to overestimate the amount of electron‐electron scattering. It is a poor defense of an approach to claim it is not so bad (or could be worse), but that is, in fact, true here. A study of the electron‐electron scattering in metals along the lines herein accounts for screening provided by other electrons but neglects the interaction of the screening electrons with each other. Including that interaction gives rise to the random‐phase approximation in which a bare Coulomb interaction is screened by the Lindhard dielectric function—but that is also an approximation as the dielectric function of the electron gas is unknown (Kukkonen and Wilkins, 1979). Kukkonen and Smith (1973) find that rather than being a factor of 5 in error (as was believed at the time of

180

KEVIN L. JENSEN

their work), the Born approximation instead overestimates by about a factor of 2 the scattering cross section and the electron‐electron contribution to the thermal resistivity. Regardless, for the present purposes, the Born approximation more than suffices to infer temperature and energy dependence of the scattering terms.

3. Electron‐Electron Scattering Any ionized impurity in a metal is quickly surrounded by the copious carriers in the conduction band. So why the interest in the ionized impurity calculation? For two reasons: first, because electron‐electron scattering dominates the other relaxation times for photoemission in metals (Tergiman et al., 1997), and second, because electron‐electron scattering can be viewed as a kind of ionized impurity scattering, albeit that the players are of equal mass and identically charged. It is in fact a difficult calculation, but the ionized impurity calculation provides guidance. The changes for electron‐electron scattering, though, are important. First, the electrons, being identical particles, both scatter into, and out of, a given state. Second, the occupation of the initial state matters, as does the final state, as an electron cannot scatter into an occupied state (a rather wordy way of saying the exclusion principle holds). The language of distribution functions is useful. The modifications for the two‐body scattering are needed, and they are to assign weighting factors of f ðkÞ ¼ probability that an electron is scattering from the state k, and ð1  f ðkÞÞ ¼ probability that an electron is scattering into the state k. The scattering event is governed by the two‐body term S ðk1 ; k2 ; k3 ; k4 Þ, in which k3 ! k1 (read as electron in state ‘‘3’’ scatters off the potential into state ‘‘1’’) and k4 ! k2 , as is symbolically indicated in k3

k4 V

k3k4 Vee k1k2 ⇒ k1

k2

The formalism for calculating the collision term then consists of associating an f factor for every arrow entering the interaction region, a factor (1 – f ) for every factor leaving the interaction region, and then integrating over all momenta. However, electrons scatter into states as well as from states. Therefore, the companion diagram in which the indices are shuffled also contributes, but with the opposite sign. The collision operator, corresponding to the diagrams suggested by the above discussion (see Wagner and Bowers, 1978, for the formal treatment) results in the interpretation of the above Feynman diagram as (Ridley, 1999; Tergiman, 1997)

181

ELECTRON EMISSION PHYSICS

ð @c f ðk3 Þ ¼ ð2pÞ9 dk1 dk2 dk4 fS ðk1 ; k2 ; k3 ; k4 Þf3 f4 ð1  f1 Þð1  f2 Þ S ðk3 ; k4 ; k1 ; k2 Þf1 f2 ð1  f3 Þð1  f4 Þg

ð481Þ

where the notation     fj  f kj ; Ej  E kj

ð482Þ

has been introduced. The principle of detailed balance indicates that when the system is in equilibrium, then the collision operator vanishes; it sets restrictions both on S and the form of the fs. First, S is symmetrical, that is, Sðk1 ; k2 ; k3 ; k4 Þ ¼ Sðk3 ; k4 ; k1 ; k2 Þ, or, alternately, the expression does not change with k1 $ k3 and k2 $ k4 . Second, when the fj’ s are replaced by their equilibrium (FD) distributions then, using the shorthand xn  bðEðkn Þ  mÞ

ð483Þ

plus the symmetry of S, Eq. (482) vanishes when 0¼

f3 f4 ð1  f1 Þð1  f2 Þ  f1 f2 ð1  f3 Þð1  f4 Þ f1 f2 f3 f4

ð484Þ

¼ ex2 þx1  ex3 þx4 when fj ¼ 1=ð1 þ exj Þ. The second line of Eq. (485) is simply a restatement of conservation of energy (the energy entering the vertex V is equal to the energy leaving). The evaluation of the relaxation time therefore makes use of departures from the FD distribution in Eq. (482) in the linearized Boltzmann’s equation. If the potential interaction is independent of spin s, and introducing the convenient but slightly obfuscating notation jki ¼ j k; si, then (recalling that jki kj i represents a Slater determinant) Sðk1 ; k2 ; k3 ; k4 Þ ¼ ¼

2p jhk1 k2 jV^ee jk3 k4 ij2 dðk1 þ k2  k3  k4 ÞdðE1 þ E2  E3  E4 Þ
h  4p 2 2 V31 þ V32  V31 V32 dðk1 þ k2  k3  k4 ÞdðE1 þ E2  E3  E4 Þ
h

ð485Þ where a sum  over spin coordinates has been performed and Vij  Vee jki  kj j . Conservation of energy and momentum are enforced

182

KEVIN L. JENSEN

in the delta functions. Analogous to Eq. (473), we have   q2 1  : Vee jki  kj j ¼ eo V k2 þ jki  kj j2 TF

ð486Þ

To say that the solution of Eq. (481) using Eqs. (482)–(486) is difficult is not an understatement, and the approach taken in most treatments is the one used here, namely, defer to the archival literature (Lawrence and Wilkins, 1973; Lugovskoy and Bray, 1998, 1999; Morel and Nozie´res, 1962; Wagner and Bowers, 1978)—there are many components—at a convenient opportunity, quote the result, and move on. However, a fair amount may be said that renders the final result plausible, and this dominates the treatment here. The focus is on the relaxation time for distributions distorted from the equilibrium distribution such that f ðkÞ  fo ðkÞ  ð@E fo ðEÞÞcðkÞ (which defines c), where the fo is understood to be FD distribution. Observe that d d fo ðEÞ ¼ ½1 þ ebðEmÞ 1 ¼ bfo f1  fo g; dE dE

ð487Þ

a function that is sharply peaked about E ¼ m [recall the discussion following Eq. (30)] indicating the magnitude of k3 approximates kF. It ‘‘follows’’ that 0 1 ððð 1 1 A 1 2p ¼@ d k1 d k2 d k4 jhk1 k2 jV^ee jk3 k4 ij2 tee ðk3 Þ 1  f3 2ð2pÞ9
h ð488Þ
f f4 ð1  f1 Þð1  f2 Þ þ f1 f2 ð1  f4 Þg  
d k1 þ k2  k3  k4 dðE1 þ E2  E3  E4 Þ where ‘‘follows’’ means that as plausible as the result appears, there is considerable effort showing it that has been passed over but can be found elsewhere (Wagner and Bowers, 1978; see also Eq. 2 in Lugovskoy and Bray, 2002, which bears greater similarity to Eq. (488)). The interpretation is that both scattering into state k1, as well as scattering from state k1, must be considered. The solution of Eq. (489) is nontrivial, but the energy delta function can be exploited to discern the behavior of interest to the present treatment insofar as the temperature and E3 dependence of tee is to be ascertained. For metals, energies a few kBT below Fermi level are filled, and so final scattering states there are precluded. The scattering electron is most probably within a kBT of the Fermi momentum, and by energy and momentum conservation, the final state is similarly constrained. That is, both jk1 j and jk3 j are comparable to kF and their difference is small—a conclusion pertaining to jk2 j and jk4 j as well, indicating that the momentum delta function

ELECTRON EMISSION PHYSICS

183

becomes a relation between the angular components rather than their magnitudes. When   screening is strong (degenerate statistics), then the term Vee jki  kj j in Eq. (487) is approximately constant. Taken together, these two observations imply the angular integrations may be handled separately from the energy integration. It is therefore sufficient for present purposes to consider the energy integral to ascertain the leading‐order temperature and energy dependence of tee(E), as the angular integrations are the source of the vexing dependence of tee on kTF —and so its evaluation is deferred to the literature. To exploit the energy delta function, switch to an energy integration via

dkn ¼

k2n dkn


 1 2m 3=2 1=2 sinydydf  m dxn sinydydf; 2b
h2

ð489Þ

where the smallness of xn (by comparison to bm over the region where the integrand is significant) has been exploited. Elsewhere [as Ziman (1985, 2001) does], it is common to write Eq. (490) in the form d kn ¼ dE n dOk

dk ¼ d 3 k  dE

dOk dOk ¼ dE  2 ; ð@E=@kÞ
h k=m

ð490Þ

where the k in the denominator is evaluated at kF; such is the origin of factors of vF ¼
hkF =m in the denominator of the coefficient of tee in the final result. The integration over E4 leaves (where the largeness of bm has been used to extend the lower limit to 1)

1

!

tee ðk3 Þ

ððð dE1 dE2 dE4 ff4 ð1  f1 Þð1  f2 Þ þ f1 f2 ð1  f4 ÞgdðE1 þ E2  E3  E4 Þ

/ ¼

1 b2

ð1 ð1 n dx1 dx2 ½ðex1 þx2 x3 þ 1Þðex1 þ 1Þðex2 þ 1Þ1 1

1

þ½ðex1 x2 þx3 þ 1Þðex1 þ 1Þðex2 þ 1Þ1

o

ð491Þ

184

KEVIN L. JENSEN

The integral may be simplified by using the relation ð1 dy c ¼ c ; xþc þ 1Þðex þ 1Þ e 1 1 ðe

ð492Þ

from which Eq. (492) can be shown to be 0 1 ð 1 1 1 ð x  x Þ ð x  x Þ 3 1 3 1 A  dx1 @ ! / 2 ½ðex1 x3 þ 1Þðex1 þ 1Þ ½ðex1 þx3 þ 1Þðex1 þ 1Þ tee ðk3 Þ b 1 ð ðex3 þ 1Þ 1 x1 dx1 x x ¼ 2 1 3 ½ ð e þ 1 Þð1  ex1 Þ b 1  1 ð493Þ ¼ p2 þ x23 2 Combining the components and hiding the angular integrations behind the newly introduced function g yields
 " 
 #
2jk3 j 1 1 kB T 2 E3  m 2 p ¼C 1þ ; ð494Þ g tee ðE3 Þ 2 E3 pkB T kTF where the collection of constants making up C and the behavior of the function g remain to be determined. Because of jhk1 k2 jV^ee jk3 k4 ij2 , it follows 2 C will resemble ðq2 =eo Þ plus a smattering of factors of p and other numbers for good measure. The function g is another matter; while its derivation is no more odious than what has transpired so far, we cite Wagner and Bowers (1978) and move on; they show 0 12 20 0 12 1 0 131 8
hKs2 @EðkÞA 4@ EðkÞ  m A Ag@2kA5 tee ðEðkÞÞ ¼ 2 1þ@ pkB T qo afs pmc2 kB T 0 1  pffiffiffiffiffiffiffiffiffiffiffiffiffi tan1 x 2 þ x2 x3 @ 1 x A pffiffiffiffiffiffiffiffiffiffiffiffiffi tan x þ gðxÞ ¼  1 þ x2 4 2 þ x2

ð495Þ

where a notation invoking the fine‐structure constant instead of the permittivity of free space is deferred to in order to make a unit analysis transparent. The sudden introduction of Ks is a not‐so‐subtle sleight‐of‐hand, as it certainly was not part of the original discussion of electron‐electron collisions. It arises as a result of additional screening by d‐band electrons at zero laser frequency, and values between 1 and 10 have been suggested for various metals (as discussed later). That implies, however, that kTF might require some changes, and so the qo factor (in a notation following Ridley) is used in

185

ELECTRON EMISSION PHYSICS

1.6

4g (x)/x3

1.2 Exact

0.8

x«1

0.4

0

x»1

−2

−1

0

1

2

3

4

5

In(x) FIGURE 64. Behavior of the angular function appearing in the electron‐electron relaxation time compared to its asymtotes.

its stead in g, where k2TF ! q2o ¼ 4kF =pKs ao :

ð496Þ

The behavior of g(x) is shown in Figure 64, along with its asymptotic limits given by 8 ð7x2 þ 12Þ > > x > > > ð15x2 þ 12Þ x3 < 0 1 gðxÞ  4> p 1 > > @1  A > > :2 x

ð x  1Þ :

ð497Þ

ð x  1Þ

The important features of Eq. (496) are the energy and temperature dependence. They are, first, that the electron‐electron relaxation time is proportional to the inverse‐square of the temperature (T 2), and second, that the energy dependence is proportional to the square of the difference of the electron energy with the Fermi level (E – m)2 when the difference is larger than the thermal energy kBT. Both follow as a natural consequence of FD statistics applied to a degenerate gas of electrons. In some treatments (Papadogiannis, Moustaizis, and Girardeau‐Montaut, 1997) and elsewhere (Jensen, 2003b), tee is parametrically represented by an equation of the form

186

KEVIN L. JENSEN

tee ¼



m h 1 2 ; A kB T

ð498Þ

where A is a dimensionless parameter of order unity. A comparison of Eqs. (495) and (498) suggests that the parameter A is approximately A  pR1 =4m; where R1 is the Rydberg energy (13.6 eV). For m ¼ 7 eV, such a comparison suggests A ¼ 1.53 if Ks ¼ 1. 4. A Sinusoidal Potential While electron‐electron scattering dominates the scattering processes that affect photoemission in metals, acoustic phonon scattering often dominates the relaxation time in thermal transport. We now turn attention to its evaluation. To prepare the way, consider the (far) simpler problem of when the perturbing potential is simply a sinusoidal (in the x direction) of the form hrjU^ep jr0 i ¼ aq eiðqxotÞ hrjr0 i;

ð499Þ

where the subscript ep (ep) reinforces that this exercise is in preparation for 0 1 scattering. It follows that treating electron‐phonon 2p Sðk; k0 Þ ¼ @ Ajaq j2 fdðEðk0 Þ  EðkÞ þ
hoÞdðk0  k þ qÞþ
h ð500Þ dðEðk0 Þ  EðkÞ 
hoÞdðk0  k  qÞg Two types of lattice vibrations occur. When the position of one vibrating lattice atom is not so far off from its neighbor, then the values of |q| are small and the dispersion relation is oðqÞ ¼ s q, where s is the sound velocity—such a mode is termed acoustic. However, if the oscillation of one atom is out of phase with its neighbors (it is ‘‘up’’ while its neighbors are ‘‘down’’), then oðqÞ  o0 ¼ a constant; it is generally observed that such vibrations can interact strongly with light, and so the mode is termed optical. That these branches occur can be intuited by considering a simple linear model (considered next). 5. Monatomic Linear Chain of Atoms Consider a 1D chain of atoms of type B (dark) and A (light) schematically illustrated below where they are joined by ‘‘springs’’ with spring constant g.

un−1

un

un+1

un+2

ELECTRON EMISSION PHYSICS

187

where the deviation from the equilibrium value is indicated by u and the atom by the subscript (e.g., n). When the atoms are displaced from their equilibrium positions, the restoring force they feel is the sum of two springs, as in the force on atom n being Fn ¼ gfðunþ1  un Þ  ðun  un1 Þg ¼ gðunþ1 þ un1  2un Þ;

ð501Þ

where g is the spring constant (the use of g in the notation—not a wonderful choice—rather than the customary k or K, is required as the latter symbols are reserved for use below). Periodicity is assumed so that the nþj atom executes the same motion as the j th atom. If the restoring forces between the atoms depend only on the magnitude of their displacement, then the atoms act as coupled harmonic oscillators, where M is the mass of the atom; in the monatomic case, the masses are the same (MB ¼ MA); in the diatomic case, they are different (and that case shall be handled separately in a subsequent section). The energy of the linear chain of atoms is therefore h2 X N 2 1 X N
E¼ kn þ g ðunþ1  un Þ2 : ð502Þ n¼1 away n¼1 replacing 2 involves 2Mstraight Quantizing Eq. (502) the kn and un with ^n and ^ operators k un , respectively, which (as introduced in Section II.C.2) ^j  ¼ idl;j . The odd notation u is persatisfy the commutation relations ½^ ul ; k haps now appreciated; u is a displacement and is a function of x—and while the commutation relations are reminiscent of those for x and k, it remains the case that u is a function of x. The basis states of the atoms are simply the product of the individual basis states, or jci ¼ ju1 iju2 i . . . juN i  ju1 u2 . . . uN i;

ð503Þ

and the Hamiltonian is
2 XN ^2 1 XN h k þ g H^ ¼ ð^unþ1  ^un Þ2 : n¼1 n n¼1 2 2M

ð504Þ

Periodicity dictates that for some integer j, ^ 1þj . . . uNþj i ¼ Eju1þj . . . uNþj i; Hju

ð505Þ

meaning that the state with j can differ from the state with j ¼ 0 by at most a phase factor ju1þj . . . uNþj i ¼ eijjn ju1 . . . uN i;

ð506Þ

where jn ¼ 2pn/N, with n an integer. The coupling of adjacent coordinates in Eq. (502) is somewhat awkward, as it does not allow for the pleasing definition of creation and annihilation operators that were gainfully used in the treatment of the harmonic oscillator

188

KEVIN L. JENSEN

when only one oscillator was present. It is therefore profitable to introduce ‘‘normalized’’ coordinates X^n and K^n defined by XN 1 XN 2pinj=N ^ e2pinj=N ; ^n ¼ p1ffiffiffiffiffi ^ X^n ¼ pffiffiffiffiffi k e ; K u j j¼1 j¼1 j N N

ð507Þ

which are immediately recognized as discrete Fourier transforms of the position and momentum operators. In an exercise of incomparable pedagogical value, it can be shown

i 

1 XN XN h ^n e2pij 0 n0 =N e2pijn=N ^ X^j ; K^j 0 ¼ ; k u n 0 n¼1 n ¼1 N 0

¼

i ð1  e2piðjj Þ Þ ¼ idj; j 0 N ð1  e2piðjj0 Þ=N Þ

ð508Þ

which shows that the normalized coordinates satisfy the requisite commutation relations sought in the treatment of the harmonic oscillator. Inserting the normalized coordinates X^n and K^n into the Hamiltonian, the relevant part of the kinetic energy is transformed to XN j¼1

XN XN XN ^2 ¼ 1 k K^ K^ 0 exp½2piðlj þ lj 0 Þ j j¼1 l¼1 l 0 ¼1 l l N XN XN ¼ K^ K^ 0 d 0 j¼1 l¼1 l l ll XN K^ K^ ¼ l¼1 l l

ð509Þ

where K^l ¼ K^l (the * denotes complex conjugation). As shown by direct substitution, X^Nn ¼ X^n , and so it is common (and shall be done below) to take the range of n to be from N=2 to þN/2 instead of 1 to N. The terms arising in the potential energy require more effort (and where the usage of the condition of periodicity in the indices is a bit more subtle): XN  j¼1

^ uj ujþ1  ^

2

¼ ¼

XN  j¼1

XN 

¼2

j¼1 XN

2^ u2j  ^ uj ^ ujþ1  ^uj ^uj1



2X^j X^j  X^j X^j e2pij=N  X^j X^j e2pij=N

j¼1



X^j X^j ½1  cosð2pj=N Þ ð510Þ

189

ELECTRON EMISSION PHYSICS

(a) 40

n = 4, N = 64

wt

30

20

10

0

0

(b) 40

10

20 30 Position [a.u.]

40

50

20 30 Position [a.u.]

40

50

20 30 Position [a.u.]

40

50

n = 8, N = 64

wt

30

20

10

0

0

(c) 40

10

n = 17, N = 64

wt

30

20

10

0

0

10

FIGURE 65. (Continues)

190

KEVIN L. JENSEN

(d) 40

n = 32, N = 64

wt

30

20

10

0

0

10

20 30 Position [a.u.]

40

50

FIGURE 65. (a) Time slices of a chain of equal masses for a low frequency; the gray area shows a representative time. Clustering of atoms results in a changing density per unit length. The line joins the regions of highest density. (b) Same as (a) but for twice the frequency; the density clusterings are closer together. (c) Same as (b) but for a high frequency; the density clusterings are very close together. The slope of the line (related to sound velocity) is larger. (d) Same as (c) but for the highest frequency; the density clustering gives way to the masses oscillating 180 out of phase with their nearest neighbors.

Using complex conjugation rather than the negative index notation, the Hamiltonian becomes ! XN=2 h2 ^ ^ 1
2 ^ ^ ^ H¼ k k þ Mon Xn Xn ; ð512Þ n¼N=2 2M n n 2 where the frequency on has been introduced and is defined by q a g g n o2n ¼ 2 ð1  cosð2pn=NÞÞ ¼ sin2 M M 2

ð513Þ

and where the wave number qn ¼ 2pn=Na and the lattice spacing a have been introduced. For the simple monatomic and isotropic system under consideration in the large N limit, the small‐angle approximation to the sine 1=2 function shows oq ¼ vsq, where vs ¼ ðga2 =4M Þ is a sound velocity. For ˚ ; vs ¼ 4910 m/s) implies example, for iron (M ¼ 0.055845 kg/mole; a ¼ 2.87 A 2 g  117 J/m . The question of dispersion, that is, the variation of on with qn beyond the small‐angle approximation, is considered below in the treatment of the two‐mass chain. It is instructive to look at the ‘‘modes’’—that is, the Hamiltonian where all of the normalized coordinates, save for one, are zero. Examples for a chain of 64 atoms are shown in Figure 65 for n ¼ 4, 8, 17, and 32. Time progresses up the vertical axis, and only a portion of the chain is shown. The gray band

ELECTRON EMISSION PHYSICS

191

represents a particular time slice, or snapshot, of the atoms’ position, and the dashed lines represent the locations of the centers of the atoms. Inspection reveals that clusters of atoms tend to form and that the center of those clusters migrates as time increases. For small n, the clusters moving in a particular direction contain a large number of atoms, wherein the direction an atom is moving is likely to be the same as its neighbors. For larger n, the clusters contain fewer atoms until n ¼ N/2, at which point the atoms oscillate around their zero point and out of phase with their immediate neighbors. The advantage of Eq. (513) is that in the normalized coordinates the Hamiltonian does not have cross terms among the Xn, so that the interpretation in terms of creation and annihilation operators, using the formalism described in Section II.A.3.b , for each index n, is possible. What is being annihilated and created are phonons, and the Hamiltonian in Eq. (513) simply counts the number of phonons with each index n and sums their energies. This can be seen explicitly by [in complete analogy with Eq. (165)] defining the operators   pffiffiffiffiffiffiffi ^n þ an X^ = 2an ^ a ¼ ik n   ^ þ an X^n =pffiffiffiffiffiffiffi ^ a{n ¼ ik 2an

ð514Þ

n

where an ¼ mon =
h Using the commutation relations in Eq. (508), it follows that 1 hon H^ ¼
2

XN=2 n¼N=2



 XN=2  1 ^ ^{n ^an þ : a{n ^ an ^ a{n þ ^ an ¼
a ho n n¼N=2 2

ð515Þ

Switching to a number representation basis, where (compare Eq. (38) and the following discussion) pffiffiffi nj1 . . . ðn  1Þ . . . Ni pffiffiffiffiffiffiffiffiffiffiffi ^ an j1 . . . n . . . Ni ¼ n þ 1j1 . . . ðn þ 1Þ . . . Ni ^ an j1 . . . n . . . Ni ¼

ð516Þ

This allows for the identification of the mean occupation number for the oscillators connected to a heat bath of temperature T to be calculated. Switch from the n to the q notation suggested by Eq. (513) in preparation for the ultimate shift to integrations over wave number (where q becomes the relevant wave vector). Then, using hnq j^ a{q ^ aq jnq i ¼ nq ; the average energy is given by

192

KEVIN L. JENSEN

0 U¼

¼

X

X

B B B ho q B
B q B @

8 X <

9 1 1= bðnq þ1=2Þ
hoq C e n þ nq : q C 2; C C X C bðnq þ1=2Þ
hoq C e nq A

ð517Þ

0

1 0 1 X 1 1 1 hoq @ boq
hoq @hnðoq Þi þ A
þ A q q 2 e 1 2

The extension from a 1D chain to the 3D continuum limit then takes Eq. (517) into Eq. (452). Such a cavalier transition from a chain of atoms to a lattice of atoms should provoke unease [for example, there may be more than one atom per unit cell, motivating the factor of r in Eq. (455)], but a systematic analysis puts such arguments on better footing (the point here, rather, is to suggest the plausibility of the transition). The point is to arrive at the investigation of a lattice specific heat, having justified the notion of the harmonic oscillator approximation and indicating why the Debye frequency is limited by the number of modes (i.e., number of atoms). To reiterate a central feature of phonons that affects the calculation of the scattering terms below: no limit as to the number of phonons in a particular state exists (that is, nj is not restricted to 0 or 1, as for fermions). The occupation factors associated with electron‐phonon scattering calculations below must account for the BE statistics of phonons. The actual mechanics are a bit involved and will be considered after the modifications associated with a linear chain with two types of atom have been more fully discussed. The use of the terminology sound velocity (how fast disturbances move in the lattice) is naturally related to the description of phonons as ‘‘acoustic’’ phonons. The picture is more complicated, however, and the subtle changes introduced by allowing adjacent atoms to be of different mass allow for a derivation of the dispersion relation in Eq. (513) by other means that help illuminate the physics. Rather than using the quantum‐mechanical model that was used previously (treated in full 3D glory by Ziman, 1985, 2001), reconsider the problem from a classical perspective, where a harmonic force €n ¼ exists between adjacent atoms. The acceleration of the nth atom x d 2 xn =dt2 is related to the forces acting on it; Eq. (501) then becomes two equations for each of the different masses, taken to be the n and (n þ 1) atoms: MA € un ¼ gðunþ1 þ un1  2un Þ MB € unþ1 ¼ gðunþ2 þ un  2unþ1 Þ

ð518Þ

193

ELECTRON EMISSION PHYSICS

with similar equations holding for other adjacent pairs of atoms. The consequences of Eq. (507) suggest that solutions of the form un ¼ CA expfiðot  nqaÞg unþ1 ¼ CB expfiðot  ðn þ 1ÞqaÞg

ð519Þ

are applicable—not surprising for harmonic oscillators. With the substitution of Eq. (519) into Eq. (518), the following equations follow 2g ðCA  CB cosðqaÞÞ MA 2g o2 CB ¼ ðCB  CA cosðqaÞÞ MB o2 CA ¼

ð520Þ

For solutions to exist, the determinant of the matrix of coefficients must vanish, and so   g  o2  2 g 2 cosðqaÞ   MA MA    0¼  g g   2 cosðqaÞ o2  2  MA MB  0 1 M þ M 4g2 A BA 2 ¼ o4  2g@ o þ sin2 ðqaÞ MA M B M A MB

ð521Þ

1.5 Optical + (r = 1) − (r = 1)

1.0

(w ±/w g)

+ (r = 0.5) − (r = 0.5) + (r = 0.25) − (r = 0.25)

0.5

Acoustic 0.0 0

30

60

90

qa [degrees] FIGURE 66. Origin of the acoustic and optical branches of a linear chain composed of two types of atoms whose masses are related by the value of r.

194

KEVIN L. JENSEN

Solutions are as follows: 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9
<
 M þ M sinðqaÞ 2 = A B 2 o ¼ g : 1 1  4MA MB M A þ MB ; MA MB :

ð522Þ

For the special case MA ¼ MB, then o2 corresponds to o2n in Eq. (513), but now another solution exists that did 2 not occur for metals in the earlier derivation. The behavior of o =og , where o2g ¼ g=M and the ‘‘reduced’’ mass M 1 ¼ MA1 þ MB1 (or half of the mass of one of the atoms when their masses are equal) is sought. Let the mass ratio MA =MB  r, then Figure 66 shows for the linear chain the dimensionless relation sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 o sinðqaÞ 2 : ð523Þ ¼ 1 1  4r 1þr og The ‘‘–’’ branch is designated acoustic and the ‘‘þ’’ is designated optical (so named because in ionic crystals, the oppositely moving atoms create a polarization that can interact with light). In monovalent metals or crystals with but one atom, the optical mode does not appear. Finally, note the near‐ linearity for small qa for the acoustic mode. In three dimensions, a similar Figure results, albeit that the acoustic mode for a 3D lattice is not as linear as implied in Figure 66. Optical phonons, by virtue of their higher frequency, are of higher energy—and their satisfying BE statistics implies that the average number of optical phonons is well below the number of acoustic phonons at generic temperatures. The particle representation of phonons allows the interaction of electrons and phonons to be treated as a particle‐particle collision, in which an electron gains or loses energy in the absorption or creation, respectively, of a phonon. If the energy of the acoustic phonon is small, then to leading order from the BE distribution in Eq. (517), it follows hnðoÞij
hokB T 

kB T
ho

ð524Þ

(where the q subscript (q) on o has been suppressed), indicating, as physical intuition suggests, that as the temperature increases, so too does the number of acoustic phonons. With more phonons, more scattering occurs during current flow, and the expectation is therefore that the resistivity of the metal will likewise increase. Eq. (524) therefore suggests that the acoustic phonon relaxation time tep (which is inversely proportional to the resistivity of the metal) should scale inversely with temperature. This holds for sufficiently high temperatures, and the effort to quantify what is meant by ‘‘sufficiently’’ is a matter of some effort, as detailed below.

ELECTRON EMISSION PHYSICS

195

6. Electron‐Phonon Scattering In terms of the normalized coordinates introduced in Eq. (507), the oscillations described by the normal modes entail variations in the local density. Consider a group of atoms in a volume Vo in their unperturbed state that, through their motion, occupy a volume V when one of their modes is stimulated. If the total mass of the crystal is M, then the change in density is given by dr ¼

M M ðV  V o Þ DV DV  ¼ M  M 2 ¼ ri ; V Vo VVo Vo Vo

ð525Þ

where ri is the number density of the crystal lattice and the term DV  V  Vo has been introduced. The fractional change in volume DV =Vo is therefore related to the fractional change in density dr=ri . Atoms moving across a surface dA, that is, x  dA, cause the density to change and so the integrated fractional change in volume can be recast as ð ð ð dr dV ¼  x  dA ¼  ð=  xÞdV ; ð526Þ ri where the last step exploits the relation between integration over surfaces to integrations over volume involving vectors (Gauss’ theorem). Eq. (526) as much as identifies dr ¼ ri =  x. This is important because the ions are slowly lumbering behemoths compared with the agile electrons, so that at all times, the electrons can be assumed to be exactly following the lattice dynamics much like flies about the plodding animals they torment; this means that whatever changes in the lattice density occur, those changes are reflected in analogous changes in the electron density. But changes in electron density can be related to changes in the electrochemical potential, and therefore, changes in the potential f under which the electrons move. Assuming that the distribution is well approximated by the FD distribution with an electrochemical potential given by mðxÞ ¼ mo þ fðxÞ, it follows (where now r refers to electron, not lattice, number density)  Ð

o dr ¼ rðxÞ  ro ¼ 2ð2pÞ3 dk fFD ðEðkÞÞ  fFD ðEðkÞÞ  Ð

o ðEðkÞÞ  2ð2pÞ3 dk fðxÞ@E fFD ð527Þ 2 3 ð 2  bf 2 b
h mk F k2  k2 5 ¼    2 k dkd4 fðxÞ p 2m F p2
h2 where the replacement of the gradient of the FD distribution with a delta function relied on the peakiness of @E f and the presumed smallness of f.

196

KEVIN L. JENSEN

On the other hand, from Poisson’s equation
 q2 q2 mkF 2 @x fðxÞ ¼ dr ¼ fðxÞ  k2TF fðxÞ; eo eo p2
h2

ð528Þ

where the Thomas–Fermi parameter k2TF ¼ ðq2 =e0 Þ@m r is familiar from Eq. (32). Thus, variations in the electron density caused by oscillations of the normal modes give rise to a potential in which the electrons move, and it follows that electron‐phonon scattering is defined by the relation by q2 ^ Þ ¼ ri V^ep ðx = ^u: eo k2TF

ð529Þ

The coefficient may have been anticipated—when electron densities are high, then a shielded Coulomb potential such as exists about the ions has more of the appearance of a Dirac delta function: q2 q2 kTF jxx0 j e ) d ð x  x0 Þ 4peo jx  x0 j eo k2TF

ð530Þ

in the strong shielding limit (recall Eqs. (470) and (471)). Although such a relation is simple, it is not the one most commonly used; more often, the sound velocity appears in the coefficient of Eq. (529) instead. Developing an expression for the sound velocity requires a detour to develop the Bohm– Staver relation. A few different methods can be used to accomplish that result. The first method is to work from the plasma frequency and observe that the electrons are tracking the motion of the ions. The plasma frequency is given by  1=2 op ¼ q2 ri =eðkÞM ;

ð531Þ

where M and ri are the mass of an ion and density of the ions, respectively, and where eo has been replaced by eðkÞ; the trick is then finding eðkÞ. A cloud of electrons around the ion potential gives rise to a ‘‘screened’’ Coulomb potential [recall the form of Eq. (470)]. From the relation UðkÞ ¼ Uext ðkÞ= eðkÞ, where U is the Fourier transform of the Coulomb potential and Uext is the unscreened potential (kTF ¼ 0), for which the Fourier transform (ignoring factors of 2p) is ð 1 q2 kTF r i k  r q2  2 e e dr¼ kTF þ k2 : ð532Þ e0 O 4pe0 r

ELECTRON EMISSION PHYSICS

197

The dispersion relation is then given by
1
 dop ðkÞ 2 q2 k2 ¼ ri 1 þ : dk 2Me0 k2TF k2TF

ð533Þ

The velocity is the k ¼ 0 limit. Charge neutrality relates the ion number density to the electron number density via a factor for the valence Z, set here to 1. It follows sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 rffiffiffiffiffiffiffiffi q2 1
hkF 2m vs ¼ ri ¼ pffiffiffi : ð534Þ ¼ 3M M 2Me0 k2TF 3 The second approach is from statistical mechanical arguments. The sound velocity is related to the compressibility w of a gas via v2s ¼ ðwmrÞ1 ;

ð535Þ

w ¼ @P lnðV Þ

ð536Þ

where

and where P, V, and r¼ N/V are the pressure, volume, and number density, respectively. Ions can be imagined as acting like massive electrons, and what is said about the latter therefore has bearing on the former. From the distribution function approach of Section I, it is known that the total energy of a gas is related by ð EN ¼ V ð2pÞ3 dkfFD ðEðkÞÞ  VhðrÞ; ð537Þ V

where the function h(r) is (temporarily) defined by this relation and the relation between chemical potential and number density. Pressure, which is defined by P ¼ @V EN , can be expressed therefore as P ¼ hðrÞ þ rh0 ðrÞ;

ð538Þ

where prime indicates derivative with respect to argument. Combining Eqs. (536) and (538), it follows that [usingV @V P ¼ r2 h00 ðrÞ]   1 ¼ r2 h00 ðrÞ ¼ V r2 @N h0 ðrÞjV ¼const : w

ð539Þ

Recalling that m is the change in energy when an additional particle is added, and from Eq. (537) that m ¼ @N EN ¼ @r hðrÞjV ¼const , it immediately follows that

198

KEVIN L. JENSEN

1 ¼ Nrð@N mÞjV ¼const w

ð540Þ

and so v2s ¼

1 2m N r @m r ; ð@N mÞjV ¼const ¼ ¼ m m 3m

ð541Þ

where the last relation follows from the zero‐temperature limit of Eq. (16). This is the sound velocity of an electron gas. The ions, however, are behaving like massive positive particles, and so the last step is to replace m with M and recover the previous result of Eq. (534); alternately, one may consider the ion background as jellium in its own right and the derivation of Eq. (541) is unchanged but for the usage of M instead of m. Consequently, the Bohm– Staver result, that the sound velocity can be expressed in terms of the chemical potential, follows. The Bohm–Staver relation allows Eq. (529) to be written q2 4 m2 1 ^ Þ ¼ ri V^ep ðx =^ u¼ =  ^u ¼ m=  ^u: 2 2 9 Mvs 3 eo kTF

ð542Þ

In other words, if the Bohm–Staver relation holds, then the deformation potential X is simply one‐third of the Fermi energy. However, this cannot be quite correct—beyond the ion valency Z (which has been ruthlessly ignored), there is the variation that must be expected in the lattice coupling constant g from material to material. Something like the Bohm–Staver relation, however, should apply, modifying the RHS of Eq. (542) by a TABLE 9 SOUND VELOCITIES OF VARIOUS METALS* Element

Atomic number

vs (exp) [m/s]

Na Mg Al Fe Cu Ag Ba W Au

11 12 13 26 29 47 56 74 79

3200 4602 6420 5950 4760 3650 1620 5174 3240

vs (BS) [m/s]

[vs(BS)/vs(exp)]2

3006 0.882 4344 0.891 5265 0.673 3576 0.361 2662 0.313 1808 0.245 1308 0.651 2515 0.236 1341 0.171 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *The Bohm–Staver relationship for sound velocity is vs ¼ 2m=3M , where m is the chemical potential and M is the mass of a lattice atom.

ELECTRON EMISSION PHYSICS

199

dimensionless constant called, say, l. Since the sound velocity is the likely culprit of differences between X and m, then l should be a ratio of energies— one being the kinetic energy of the lattice and the remaining energy being the Fermi energy, as deduced from the coefficient of Eq. (542). Therefore, let 4 ^Þ ) lm= ^ u V^ep ðx 3 Z m l 3 Mv2s

ð543Þ

where l ¼ 1/2 if the Bohm–Staver relation holds. Table 9 compares the commonly accepted sound velocity with the sound velocity predicted by the Bohm–Staver relation for several metals, along with the values of lexp, where ‘‘exp’’ indicates the empirical value. A ‘‘good’’ metal like sodium has reasonably good agreement, whereas the agreement with the transition metals is spottier. The time is now ripe to switch to the canonical variables X and K that describe the phonons, but difficulties immediately present themselves. First, u is a real quantity, and so the urge to make a trivial multidimensional generalization of Eq. (507) is ill advised. Second, the potential is the divergence of u, but that is unlike the construction leading to the creation and annihilation operators as done for the 1D case in Eq. (514). More care is indicated, and in fact, much more care is required. It can be found elsewhere (e.g., Wagner and Bowers, 1978); here, a brief sketch must suffice. If u describes a wave phenomenon, then it vanishes when acted on by the D’Alambertian operator □  =2  c2 d2t ; that is, □u ¼ 0, where the velocity c is replaced here by vs (see Chapter 10 of Goldstein, 1980). Equation (543) does not entail =2u but rather it would give rise to =ð=  u Þ, and if □u ¼ 0 is to hold, then several consequences result. First, using the relation =2 uðx; tÞ ¼ =ð=  xÞ  =
ð=
xÞ

ð544Þ

indicates the second term must vanish if Eq. (543) is to be exploited. Second, the relation uðx; tÞ ¼

! 1X Xk eiðkxo ktÞ k>0 V

ð545Þ

subsequently entails that k k Xk . Both observations entail that shear and vorticity do not occur, and the k > 0 condition on the summation in Eq. (545) implies that an overall movement (translation) of the medium is foresworn. In practice, these conditions mean that the proper generalization of the 1D problem to 3D gives

200

KEVIN L. JENSEN

sffiffiffiffiffiffiffiffiffiffiffiffiffi o h n iok t
^ Xk ¼ þ ^a{k eiok t ; ak e 2Mok from which it follows sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o X h
^ ^ ak eiðkxok tÞ þ ^a{k eiðkxok tÞ ; uðx; tÞ ¼ 2Mok ri V k>0

ð546Þ

ð547Þ

where the direction of u is understood to be in the direction of k; the notation could be better (but is not). Finally, note that ri is the ion number density, so that riV ¼ N ¼ the number of ions, which harkens back to the definitions introduced in Eq. (507). With Eq. (547), much follows in short order. First and foremost is the electron‐phonon potential interaction from Eq. (543), or 4 ^Þ ¼ lm=  ^ u V^ep ðx 3 o 1 Xn ¼ pffiffiffiffi ak ^ ak eiðkxok tÞ þ a k ^ a{k eiðkxok tÞ ; V k>0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ho k
ak  i lm 3 2Mv2s ri

ð548Þ

hok =3r when l ¼ 1/2. The coefficient of the divergence of u where jak j2 ¼ m
has units of energy; the form here of V^ep ¼ ð4=3Þlm=  ^u is not the one typically encountered in practice. Elsewhere (e.g., Ridley, 1999) it is instead written V^ep ¼ w=  ^ u, and X is known as the deformation potential. This shall be returned to in due time, but it is worth noting here that semiconductors, unlike the metals so far considered, have different properties along different crystal axes, one example being effective mass and another being that the deformation potential exhibits elastic anisotropy. The matter of the creation and annihilation operators must now be considered, which in turn means greater attention to the initial and final states. The approach here will be to consider one of them, then using Feynman diagram–like arguments, to infer the remainder. The relaxation time involves the evaluation of a term resembling jh f jU^ep jiij2 , but what constitutes the initial and final states needs examination. Clearly, there are momentum states jki in the initial state jii, but by virtue of the harmonic oscillator creation and annihilation operators in Eq. (548), there are also phonon states jni. This leads to identifying jii ) jkijni, where the first ket refers to the momentum states of the electrons and the second to the occupation number of the phonons. Consider now what is meant by the first term in

201

ELECTRON EMISSION PHYSICS

1 − f2

1 − f2 n(w ) + 1

S(k2,k1) n(w )

f1

S(k2,k1) f1

S(k2,k1)f1(1−f2)n(w q)

S(k2,k1)f1(1−f2){n(w q) + 1}

FIGURE 67. Feynman diagrams for the phonon interaction terms.

Eq. (548) sandwiched between the initial and final states in the circumstance when the initial state is acted on by an annihilation operator. It follows h f jak eikx ^ ak jn1 i ak jii ¼ hk2 jak eikx jk1 ihn2 j^ pffiffiffiffiffiffiffiffiffiffiffiffi ÐÐ ¼ dx1 dx2 hk2 jx2 ihx2 jak ei kx jx1 ihx1 jk1 i nðok Þ pffiffiffiffiffiffiffiffiffiffiffiffi Ð ¼ dx1 ak eiðkþk1 k2 Þx nðok Þ pffiffiffiffiffiffiffiffiffiffiffiffi ¼ ak dðk2  k1  kÞ nðok Þ

ð549Þ

The delta function ensures that jak j ) jak2 k1 j. In particular, S ðk2 ; k1 Þ ¼

  2p jak2 k1 j2 d E2  E1
ho ðk2 k1 Þ : h


ð550Þ

In finding the collision integral, the occupation of the initial state (given by f1) and the vacancy of the final state (given by 1 – f2) figure in a form that resembles ð   dk1 S ðk2 ; k1 Þf1 ð1  f2 Þn oq ; ð551Þ where q  k2  k1 for the subscript on o and the sign subscript on S refers to the sign of k in the delta function embedded within S. There is a diagrammatic interpretation of integrals of the form in Eq. (551): straight lines refer to electrons and carry factors of f going into a vertex, or factors of1 – f when leaving a vertex; wavy lines refer to phonons and carry factors of n oq when  going into a vertex or factors of n oq þ 1 when leaving a vertex; and the vertex itself carries a factor of S. An overall conservation of momentum delta function is buried in S and originates in Eq. (551), whereas an overall conservation of energy delta function from Fermi’s golden rule is appended at the end. In Feynman‐diagram parlance, this amounts to the Feynman diagrams represented in Figure 67: q is the difference between the entering and exiting momenta, the electrons are straight lines, the phonons are wiggly

202

KEVIN L. JENSEN

lines, and the interaction is the center circle. There are a total of four such diagrams: two with entering phonons, two with exiting, and a shuffling of the Fermi lines. In the example shown in Figure 67, the absorption case is the Figure on the left; the emission case is the Figure on the right. A careful collection of terms then shows that the collision integral is of the form ð 3 @c f ðk2 Þ ¼ ð2pÞ dk1 fSþ ðk1 ; k2 Þ½ f2 ð1  f1 Þð1 þ n21 Þ  f1 ð1  f2 Þn21  þS ðk1 ; k2 Þ½ f2 ð1  f1 Þn12  f1 ð1  f2 Þð1 þ n12 Þg ð552Þ where the shorthand of Eq. (482) is used, a new shorthand nk1 k2  n12 is introduced, and all other gremlins of notation can be inferred. Reducing this equation to a simpler form takes additional work. In writing Eq. (552), it was tacitly assumed that the S factors did not depend on the order of the vector arguments. As natural as that may naı¨vely seem, it is not generally true, but rather the case only for potentials of a general type—namely, those unaffected by time reversal and inversion—but showing why that means that S is insensitive to the order of its arguments is rather complicated. Intuitively, it is known that time reversal does not affect position but reverses the sign on momentum, as the velocity
hk=m ¼ dx=dt changes sign because dt changes sign, and that observing an interaction with the clock running backward switches the entering and exiting momenta. Neither intuition nor belief is proof: proof demands more. Invariance under spatial inversion (‘‘parity’’) focuses on the consequences of an operator ^up that behaves according to   U^p xi ¼   xi     ð553Þ hk^ up xi ¼ hk  xi ¼ hkxi where the second line is a consequence of the 3D generalization of Eq. (38). ^ where I is the Similarly, it is obvious by repeated application that ^u2p ¼ I, identity operator. Recall that an operator that is a constant of the motion commutes with the Hamiltonian, implying that ^ 1 ¼ H: ^ U^pHU p

ð554Þ

By inspection the kinetic energy operator is unaffected by ^up because *  + *  + 2 ^2  2 2 ^2  

h2 k k h h
k 0 0 0  k ¼ k : dðk  k Þ ¼ k k ð555Þ 2m  2m 2m 

ELECTRON EMISSION PHYSICS

203

Therefore the Hamiltonian’s invariance with respect to inversions requires an examination of the potential term V^ep . Using Eq. (549), it follows that     hxV^ep x0 i ¼ hxU^p{V^epU^p x0 i: ð556Þ The LHS is

ð       hxV^ep x0 i ¼ dkdk0 hxkihkV^ep k0 ihk0 x0 i ð    0 0 0 ^q k ihk x i / dkdk0 hxkihk

ð557Þ

2  From Eq. (557), a factor of akk0  appears. The RHS of Eq. (556) is slightly more work and yields    0 0   Ð ^q k ihk   x0 i hxU^p{V^ep ^ up x0 i / dkdk0 hxkihk ð558Þ     Ð ^q   k0 ihk0 x0 i / dkdk0 hxkihk and therefore yields a factor of jakþk0 j2 . Invariance with respect to inversion therefore allows us to conclude Sðk; k0 Þ ¼ S ðk;k0 Þ:

ð559Þ

Time reversal is slightly more involved. If a wave function exists (assume c is an energy eigenstate) such that the actions of time reversal result in cðx; tÞ ) cðx; tÞ, then the operator formalism contains some interesting differences. Observe that c ðx; tÞ satisfies the same (Schro¨dinger) equation that cðx; tÞ does, and therefore the time‐reversal operator entails complex conjugation as per Tcðx; tÞ ¼ c ðx; tÞ. In bra‐ket notation, given that    cðtÞi ¼ exp iHt=
^ h cð0Þi; ð560Þ it follows that

      hxT^cðtÞi ¼ hxt^cðtÞi ¼ hxcðtÞi ¼ hcðtÞxi;

ð561Þ

where the action of the lower‐case t^ operator is to simply change t to –t. Time reversal T^ therefore entails the actions of t^ and complex conjugation; the latter observation makes incoming particles appear to be outgoing when time is reversed. We have ð       2    ^ hcf ðtÞ ci ðtÞi ) hcf T ci i ¼ dxhcf ðtÞT^xihxT^ci ðtÞi ð ð562Þ    ¼ dxhcf ðtÞxi hxci ðtÞi ¼ hci ðtÞcf ðtÞi  Ð  where the facts that T^ is its own inverse and dx xihx ¼ I^ (apart from factors of 2p) have been used. Reversal of time therefore entails that the

204

KEVIN L. JENSEN

initial and final configurations switch, as intuitively expected but now proven. The analog of Eq. (556) must now be checked (the invariance of the kinetic energy operator under the action of T being trivially shown):     hcf ðtÞT^V^epT^1 ci ðtÞi ¼ hci ðtÞV^ep cf ðtÞi: ð563Þ To determine the impact of time reversal on S, then, the action of T^ on the momentum kets requires consideration. A short analysis shows ð      hkT^cðtÞi ¼ dxhkxihxT^cðtÞi ð ð564Þ    ¼ dxhx  kihcðtÞxi ¼ hcðtÞ  ki after which it is straightforward to demonstrate that

ð       ^q   kihkcf ðtÞi: hcf ðtÞT^V^epT^ 1 ci ðtÞi ¼ dkdk0 hci ðtÞk0 ihk0 

ð565Þ

Invariance to time reversal therefore entails Sðk; k0 Þ ¼ Sðk0 ; kÞ

ð566Þ

(a related consequence is that oq ¼ oq ). Consequently, the combined action of Eqs. (559) and (566) shows that if the potential is both time and inversion invariant, then Sðk; k0 Þ ¼ Sðk0 ; kÞ;

ð567Þ

a result that, however intuitive, did require some effort to show. Less obvious but soon to be appreciated is that the outcome allows substantial simplification of the equation for electron‐phonon scattering, and so we now return to the consideration of Eq. (552). It is seen immediately that Eq. (567) entails that @c f ðk2 Þ ¼ 0 when the distributions are replaced by equilibrium FD distributions—as it should. After all, this is what is meant by equilibrium—that as many particles scatter into as out of a particular state (and why Eq. (567) was intuitively expected). Now consider deviations from equilibrium, so that f ðk2 Þ ¼ f ðEðk2 ÞÞ þ dk1 ;k2 df ðk1 Þ ¼ f2 þ dk1 ;k2 df ;

ð568Þ

for which @c f ðk2 Þ ) df =tðk2 Þ and the RHS of Eq. (552) becomes more complicated. The following observation(s) help: if fj are FD, and n BE distributions, then using the notation xj  b Eðkj Þ  m (for a moment) and  1 fj ¼ ð1 þ e xj Þ1 ; nij ¼ exp½b
hoki kj   1 ; ð569Þ it can be shown that (it is emphasized that subscripts in the compact notation

205

ELECTRON EMISSION PHYSICS

nij have a different meaning than they do for fj) f1  f2 ¼ 

ð1  f 2 Þ f 1 n12

¼þ

ð1  f 1 Þ f 2 n21

ð570Þ

where either the top or bottom relation holds. The top will be shown explicitly (and the proof of the bottom left to independent confirmation): f1  f2 ¼

e x2  e x1 ð1  ex1 x2 Þ ¼  ; ð 1 þ e x1 Þ ð 1 þ e x2 Þ ð1 þ ex1 Þð1 þ ex2 Þ

ð571Þ

where the final transition, from Eq. (571) to Eq. (570), involves resubstituting Eq. (569). It is straightforward, albeit requiring care, to show that with the substitution f1 ) f1 þ df and all fj are then taken to be FD distributions, it follows f2 ð1  ð f1 þ df ÞÞð1 þ n21 Þ  ð f1 þ df Þð1  f2 Þn21 ¼ df f f2 þ n21 g f2 ð1  ðf1 þ df ÞÞn12  ðf1 þ df Þð1  f2 Þð1 þ n12 Þ ¼ df f f2  n12  1g ð572Þ which is as it should be; the terms independent of df cancel for FD statistics. Insertion of Eq. (572) into Eq. (552) produces ð 1 2p 1 dk1 jak2 k1 j2 fðf1 þ n21 ÞdðE2  E1 þ
hok2 k1 Þ ¼ tep ðk2 Þ h ð2pÞ3
hok2 k1 Þg ð573Þ þð1 þ n21  f1 ÞdðE2  E1 
where use has been made of oq ¼ oq and n2–1 ¼ n1–2. It is profitable to switch from k1 as the integration term to q ¼ k1  k2 , for which (where the expanded notation has been reverted to) ð

       1 2p 1 dqjaq j2 fo E2 þ
hoq þ n
¼ h oq hoq d E2  E1 ðqÞ þ
tep ðk2 Þ h ð2pÞ3


       þ 1þn
hoq  fo E2 
hoq d E2  E1 ðqÞ 
hoq g

ð574Þ

where the substitutions made in the FD functions are allowed by the arguments of the Dirac delta functions. The arguments of the delta functions appear to obfuscate matters, but on closer examination, we see that

206

KEVIN L. JENSEN

E2  E1 ðqÞ
ho q ¼


2  2 h q þ 2k  q
hoq 2m


2 k h vs oq cosy
hoq m 0 1 v k ¼
ho@ cosy 1A vs 

ð575Þ

here vk ¼
hk=m is the electron velocity and q2 is taken to be negligible. The argument of the delta functions, therefore, concerns the angle between k and q, rather than the magnitude of q. The angular integration can therefore be done separately from the q integration. Using dðaxÞdx ¼ dðyÞdy=jaj (where x, y, and a are dummy terms), Eq. (548) to find jaq j2 , using dq ¼ 2pq2 dqdx, recalling that vs ¼ o=q, suppressing the q subscript on o (because o q ¼ ojqj , there is no further utility in retaining it), collecting terms, and using Eq. (548), then Eq. (574) becomes 0 1 ð 2 2
1 1 p h vs 3 @ A ¼ ð2pÞvs o2 doð
l hoÞ1
tep ðkÞ 4p2
h kF m vk 8 0 9 ð576Þ 1 0 1 ð1 < = v v s s dx d@x  A½1 þ n  fo  þ d@x þ A½n þ foþ  : ; vk vk 1 where E ¼ E(k), n ¼ nð
hoÞ, and fo ¼ fo ðE
hoÞ. Collecting terms and noting that the second integral is simply the product of the factor in square brackets with Yðvk  vs Þ results in ð 1 2lp ¼ o2 JðE; oÞdoYðvk  vs Þ tðkÞ ð2kF vs Þ2 ð577Þ JðE; oÞ  ½1 þ 2nð
hoÞ  fo ðE 
hoÞ þ fo ðE þ
hoÞ which is fundamentally the same as the form given in Wagner and Bowers (1978). We restrict attention to those cases in which the electron velocity exceeds the sound velocity to dispense with the Heaviside step function. Using the definitions of the BE and FD distributions, it is a straightforward exercise to show that JðE; oÞ ¼ ¼

ðexo þ 1ÞðexE þ 1Þ2 ðexE þxo þ 1ÞðexE xo þ 1Þðexo  1Þ fcoshðxE Þ þ 1gfcoshðxo Þ þ 1g fcoshðxE Þ þ coshðxo Þgsinhðxo Þ

ð578Þ

ELECTRON EMISSION PHYSICS

207

where the shorthand xE ¼ bðE  mÞ and xo ¼ b
ho has been used, and where the second line is intentionally reminiscent of Eq. (15) in Gasparov and Huguenin (1993), whereas the formulation in the third line makes better use of the inherent symmetry and asymptotes. Several consequences immediately follow that are of interest here, and they relate to the temperature dependence of the scattering rate at the Fermi level and the energy dependence for low temperatures. We deal with the temperature dependence at the Fermi level first. For E ðkÞ  m, then Jðm; oÞ ¼ 2=sinhðb
hoÞ. It follows that ð oD 1 lp o2 ¼ do 2 tep ðkF Þ ðkF vs Þ 0 sinhðb
ho Þ ð xD ð579Þ lp x2 3 ¼ dx ð k T Þ B
hð
hkF vs Þ2 0 sinhðxÞ where xD ¼ b
hoD ¼ TD =T and where oD and TD ¼
hoD =kB are the Debye frequency and temperature, respectively. The upper limit of the integral assumes that oD /vs is smaller than kF; if this were not the case, then the upper limit of the integral would have to be xF ¼ b
hvs kF . For large xD, the integral in the second line is 7zð3Þ=2, whereas for small xD, it becomes x2D =2, which indicates that for low temperatures, tep ðxD  1Þ / T 3 , but for high temperatures, tep ðxD  1Þ / T 1 . At room temperature for rather generic parameters, the relaxation time is tens of femtoseconds. Finally, the units of (kFvs) are [fs1], which offsets the units of o in the integrand— an observation useful when considering higher‐order powers of o to be encountered in the momentum relaxation time below. The high temperature relation is an oft‐used and well‐known result (Jensen, Feldman, Moody, and O’Shea, 2006a; Papadogiannis, Moustaizis, and Girardeau‐Montaut, 1997), a commonly used form being given by

h 1 tep ðk ¼ kF Þ  ; 2pl0 kB T

ð580Þ

where the ‘‘0’’ on l reiterates that it is different than the l used in Eq. (579) by a matter of a few constant factors—in particular, for xD « 1, then lo ¼ l/4. The low‐temperature limit implied by Eq. (579) needs modification for the momentum relaxation time (which has bearing on the resistivity), however, where a factor of 1  cosy must be inserted into the integrand. The relationship 1  cosy ¼ 2sin2 ðy=2Þ;

ð581Þ

208

KEVIN L. JENSEN

where y is the angle between the initial and final momentum states, plus the relation between the initial and final states at the Fermi level given by
 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y 2 q¼ k1  k2  2kF ð1  cosyÞ ¼ 2kF sin 2

ð582Þ

allows the factor in Eq. (581), when put into the relaxation time as per Eq. (468), to result in an additional factor of o2 in the integrand, and therefore, an additional factor of T2 in Eq. (579). The resulting low‐ temperature dependence of the relaxation time as tep ðk ¼ kF Þ / T 5 is additionally well known and referred to as Bloch–Gru¨neisen behavior. The proper limits have now been ascertained. As a historical matter, a detailed exposition of the Bloch–Gru¨neisen formulation can be found in Wilson (1938), who observes with unflappable British understatement, that ‘‘(t)he formula we have just derived is so complicated that we must make some approximations before going any further.’’ Wilson, after making the promised and reasonable approximations, arrives at the conclusion that the relaxation time is (a variation of his Eq. (18)) 0 15 20 13 3 ð 1 8pl0 kB TD @ T A TD =T x5 T  dx þ O4@ A5 x  1Þð1  ex Þ tep ðkF Þ TD T
h ð e D 0 1 0 15 0 8pl0 kB TD @ T A @ TD A ¼ W 5; TD
h T

ð583Þ

The second term of the first line (the one we labor to ignore) corresponds to the temperature equivalent of the minimum energy necessary to excite s‐d transitions and is not considered further. The coefficient containing l0 is made to converge with the common high‐temperature representation of te‐p. In the literature, W–(n,x) is designated as Jn(x) and referred to as the Bloch–Gru¨neisen function (Fuller, 1974; Westlake and Alfred, 1968; White and Woods, 1959). Note also that the idealized l of the Bohm–Staver relation has been replaced with l0. Although a relation between the two was suggested previously (specifically, in Eq. (580) and the discussion following it), any connection in the future is casually ignored, as l0, used to generalize from the Bohm–Staver relation but not otherwise specified, must be pressured to take on whatever constants that additionally accrue in the transition from Eq. (579) to (583); for our present purposes, it is sufficient to know that its value is assigned below using comparisons to experimental data.

209

ELECTRON EMISSION PHYSICS

(a)

Resistivity [mΩ-cm]

101

100 Cu Cu (BG) Ag Ag (BG) Au Au (BG)

10−1 10−2

Pb Pb (BG) Mg Mg (BG) W W (BG)

100

1000

Temperature [K] (b) 2

In{r (T)/r(TD)}

0 −2 −4

BG

Cu

Na

−6

Ag

Mg

Fe

−8

Au

W

Al

0.1

1

10

T/TD

(c) 500

Debye temperature [K]

400

300

200 Kittel 100

Least squares Increasing atomic number

0.0

Na Mg Al Fe

Ni Cu Mo Ag W

Element FIGURE 68. (Continues)

Pt Au Pb

210

KEVIN L. JENSEN

100

W-(5,x)

10−1 10−2 Numerical 10−3

120 z(5) x5 (x/4)/{1 + (18x2)−1}

10−4

Hybrid 10−5

−3

−2

−1 In(x)

0

1

FIGURE 69. Behavior of the Block–Gru¨neisen function W–(5,x) (calculated numerically) compared to its asymptotic limits and the ‘‘hybrid’’ approximation.

The performance of Eq. (583), caveats and all, is rather spectacular. If acoustic phonon scattering dominates (as it does at the Fermi level), then the ratio of the scattering rate at temperature T to that at temperature TD will be the same as the ratio of the resistivities for the same temperatures. Consequently, metals should exhibit a universal curve when t(T)/t(TD) is plotted as a function of T/TD—given that the resistivity for metals is dominated by acoustic phonon scattering, it follows that the ratio of the resistivities as a function of T/TD should likewise exhibit a universal characteristic of
rðT; TD Þ ¼ rðTo ; TD Þ

T To

5

W ð5; TD =T Þ ; W ð5; TD =To Þ

ð584Þ

where To is a suitably chosen reference temperature. An example of the performance of Eq. (584) for a variety of metals is shown in Figure 68, where To is the highest temperature for which a resistivity value is available, the ‘‘hybrid’’ approximation to W–(5,x) is used (see Eq. (A7) and Figure 69), the resistivity values have been taken from CRC tables (Weast, 1988), and the Debye temperatures are from Table 1 of Kittel (1996). The agreement is astounding and can be repeated for other metals not shown. The values of FIGURE 68. (a) Resistivity of select metals compared to the predictions of Eq. (584). (b) Demonstration that a wide variety of metal resistivities are well described by a Bloch–Gru¨neisen function dependence [Eq. (584)] using the least‐squares estimate for the Debye temperature. (c) Debye temperatures evaluated from a least‐squares estimate on the presumption that metals follow a Bloch–Gru¨neisen behavior as shown in (b). A comparison to the Debye temperatures given by Kittel (1962) is also shown.

211

ELECTRON EMISSION PHYSICS TABLE 10 DEBYE TEMPERATURES* Atomic number

Element

TD (Kittel)

r(TD)

LS‐Fit

r(TD)‐fit

11 12 13 26 28 29 42 47 74 78 79 82

Na Mg Al Fe Ni Cu Mo Ag W Pt Au Pb

158.00 400.00 428.00 462.45 450.00 343.00 450.00 225.00 400.00 240.00 165.00 105.00

4.9300 14.400 10.180 20.662 38.600 6.0410 21.200 5.6400 21.500 32.000 7.8600 38.300

200.66 313.77 399.54 470.00 434.91 274.61 370.24 217.40 361.57 187.93 123.52 82.180

2.9018 4.7178 3.8648 57.100 13.652 1.5538 7.2660 1.1334 6.9304 6.3008 0.84522 6.7215

*‘‘Kittel’’ refers to Table 1 of Kittel (1996), whereas ‘‘LS‐Fit’’ refers to a least‐squares fit of the ratio of the resistivity to its value at the Debye temperature as predicted by the Bloch– Gru¨nesien theory in which the Debye temperature is treated as an adjustable parameter. Resistivity values were from the CRC tables (Weast, 1988); the fitting excluded the 1 K and 10 K resistivity values.

the Debye temperature for various solids depend on temperature range and sample purity (White and Woods, 1959). Therefore, if the Debye temperature is treated as an adjustable parameter in fitting Eq. (584) to actual resistivity data, the universal feature of the behavior of metals follows the relation implied by Eq. (584) quite well: if the reference temperature To is taken to be the Debye temperature TD, then TD becomes a parameter that can be extrapolated by performing a least‐squares comparison of tabulated resisitivity with a Bloch–Gru¨neisen function such that

2  1 @ X rðTn Þ  5 W ð5; x Þ ln  ln x  n  @To rðTo Þ W ð5; 1Þ

¼0

ð585Þ

To ¼TD

is minimized when To ¼ TD, where n is an index of the tabulated values of the resistivity as a function of temperature and W (n,x) is defined in Eq. (A2) with approximations in Eq. (A7). Note that r(T)/r(TD) as a function of T/TD is implied by Eq. (584) to be a universal relation for all metals. Performing the minimization using CRC resistivity data (for which the lowest temperature values, typically 1 K and 10 K, are excluded) results in Figure 68b (which shows greater correlation than Figure 11 of [White and Woods, 1959] and thereby indirectly points out the sensitivity on the estimates of the Debye temperature). Finally, the fitted values of the Debye temperature implied by a

212

KEVIN L. JENSEN

fit to Eq. (585) are in fact often quite close to tabulated values obtained via other means (Figure 68c): comparison of the fitted Debye temperatures with values quoted in Kittel is given explicitly in Table 10. For electron energies in excess of the Fermi level, refer to Eqs. (577) and (578) to consider the frequency dependence of JðE; oÞ. The presence of an o2 in the integrand of Eq. (579) (or the higher powers of o for the momentum relaxation time) offsets the sinhðxo Þ in the denominator when the argument is small, and so the small‐case o can be ignored. At low temperatures, xo and xE are large because of b, and so JðE; oÞjb1 

coshðxE Þ fcoshðxE Þ þ coshðxo Þg

ð586Þ

in the regions for which the integrand is significant. It is seen, therefore, that at low temperatures, JðE; oÞ behaves analogously to a Heaviside step function, and so ð oD 1 lp ¼ o2 JðE; oÞdo tep ðkF Þ ðkF vs Þ2 0 ð oD ð587Þ lp o2  do ðkF vs Þ2 0 fexp½bðjE  mj 
hoÞ þ 1g where the power of o is increased by 2 if the momentum relaxation time is considered as per Eq. (582). The absolute value of E – m is a consequence of the behavior of coshðxÞ  ejxj =2 when the magnitude of x is large. To leading order, then, it follows that if oD > jE  mj, then the integral will be proportional to ðE  mÞ3 , whereas if oD < jE  mj, then the integral is a constant, that is, for large b ð oD o2 jE  mj3 jE  mj <
hoD do  3
h3 ð588Þ hoÞ þ 1g ð
hoD Þ3 jE  mj >
hoD 0 fexp½bðE  m 
whereas for the momentum relaxation time, the power is increased by 2. For general conditions and a generic metal with a Debye temperature of 350 K (similar to copper), then
hoD  0:03 eV and for all practical purposes for typical photoemission wavelengths that are to be considered below, the acoustic relaxation time can be taken as approximately constant. That conclusion is not a priori so for semiconductors, but that is another, and much longer, story. It was suggested previously that the value of l would be specified by empirical data rather than just taken as 1/2. That is not quite accurate; what shall be adjusted instead is the value of the deformation potential. The high‐temperature asymptote of the acoustic phonon relaxation time

213

Resistivity [mΩ-cm]

ELECTRON EMISSION PHYSICS

10−1

Cu Ag Au Ni Mg

10−2

10−3 0

20 40 Temperature [kelvin]

60

FIGURE 70. Low‐temperature behavior of the resistivity shows that the impact of the scattering of defects persists at low temperatures.

entailed by Eq. (580) can be written in the form (Ridley, 1999) tep ¼

2
h pMri
h3 v2s ¼ ; pl0 kB T mkF 2 kB T

ð589Þ

where the two equivalent formulations are given. This implies that l0 jBohmStaver ¼

92 2m2

ð590Þ

as long as the ion number density and electron number density are the same, and the sound velocity is given by the Bohm–Staver relation ð1=2ÞMv2s ¼ m=3. If so, l ¼ 1/2 implies X ¼ m/3, as before. However, Mri is the mass density, the sound velocity is an empirical quantity, and so specifying l by empirical relations is tantamount to finding appropriate values of X. 7. Matthiesen’s Rule and the Specification of Scattering Terms The low‐temperature behavior of the resistivity was not shown in Figure 68. According to the Bloch–Gru¨neisen relation entailed in Eq. (584), the resistivity should approach 0 as T5—but in fact, as shown in Figure 70 for another sampling of metals, what is seen, rather, is that the resistivity tapers off to what appears to be a finite value as the temperature becomes cryogenic. This residual resistivity is due to defects in the lattice that serve as scattering centers to electron transport (and the reason why the low‐temperature resisitivity values were truncated in the least‐squares estimation of the Debye temperature). If scattering mechanisms are independent, then the probability of scattering should approximately follow the sum of the individual

214

KEVIN L. JENSEN

ln(t exp [fs])

12

Cu Ag Au Pb W

8

4

0 0

2

4 ln(T [K])

6

FIGURE 71. The total relaxation time as inferred from thermal conductivity data.

scattering probabilities; that is, the sum of the inverse relaxation times, and therefore, the total resistivity should simply be the sum of the partial resisitivities, a relation that is empirically supported and known as Matthiesen’s rule, and given by 1 1 1 1 ¼ þ þ ttotal tdlf tee tep

ð591Þ

Experimental relations (Kanter, 1970) tend to consider the closely related mean free path that follows the same relation, and in addition pay attention to complications not considered here (DOS and interaction with d electrons, for example), but these complications are outside the present scope. Relaxation times, being related to thermal conductivities via Eq. (439), allow for an indirect method to determine how well a prescription like Eq. (591) applies in practice. By knowing the tabulated values of thermal conductivity as a function of temperature coupled with the specific heat evaluated at room temperature and the relation entailed in Eq. (441), it is possible to recast the thermal conductivity data as relaxation time data (Jensen, Feldman, Moody, and O’Shea, 2006a) by using Eq. (439), or t ð mÞ 

3m kexp ðT Þ ; 2m gexp T

ð592Þ

where the exp subscript (exp) reinforces that values for these quantities are obtained from tabulated data in the literature [e.g., Gray (1972) or CRC tables]. Figure 71 shows such application for a variety of metals. Several

ELECTRON EMISSION PHYSICS

215

features are evident. For the metals shown (and for metals in general), the relaxation time tends to go to a constant value, presumably dictated by tdef as the temperature drops—note that lead (Pb) does not quite match this, because lead becomes a superconductor at temperatures below 7.2 K. In addition, there is a change in slope for the higher temperatures, primarily corresponding to the change in behavior of the e‐p relaxation time as a function of temperature: at the higher temperatures, the slope approaches (1) as expected. Generally, the e‐e relaxation time is dominated by the others; thus, not much can be said regarding it for now—but that changes when the energy of the electron is higher than the Fermi level as a consequence of photoexcitation. The matching of thermal conductivity data (or rather, to the relaxation time inferred from thermal conductivity data) requires that values of three quantities that have not been heretofore given be provided as follows:  The value of tdef, which can be obtained from low‐temperature thermal

conductivity data

 The value of Ks in the electron‐electron scattering rate, which shall be

obtained by comparisons to more comprehensive theory and simulation than the treatment given here—where available; and finally  The value of X, which shall be inferred by a best‐fit model to the total relaxation time inferred from thermal conductivity. For a number of metals that are of interest, the requisite detailed studies and simulations may not be available, and in those cases, a combination of arguing by analogy and a minimization of least‐squares differences is used. While it is true that a number of approximations have been folded seamlessly into the exposition, perhaps the most consequential ones are the parabolic energy‐momentum relation and a general ignoring of the shape of the Fermi surface and the DOS (which are not trivial). Nevertheless, for present purposes, the specification of the three bulleted quantities above provides more than adequate agreement that can be used to account for the role of scattering in the photoemission process and to allow for the usage of an idealized model of the electron distribution. Consider next the electron‐electron relaxation time, which has been the subject of sustained interest in both theory and simulation (Campillo et al., 1999; Krolikowski and Spicer, 1969; Ladstadter et al., 2004; Lugovskoy and Bray, 1998; Quinn, 1962; Wertheim et al., 1992). By comparing the theoretical formula entailed in Eq. (495) to the Monte Carlo simulations and detailed calculations, it is possible to form estimates of Ks. Consider the examples shown in Figure 72, where Eq. (495) is compared to theoretical calculations of electron scattering for gold and copper by Ladstadter et al.

216

KEVIN L. JENSEN

(a) Cu Ks(eff) = 1.684

tee [fs]

100

10 Ladstadter Lugovskoy Ks = 1 Lugovskoy Ks= 5.2 Theory

1

0

1

2 3 E - m [eV]

4

5

(b) Ag Ks(eff) = 14.417

tee [fs]

100

10 Kanter Krolikowski Theory 1

0

1

2 E-m [eV]

3

4

(c) Au Ks(eff) = 6.4546

tee [fs]

100

10 Ladstadter Theory 1

0

1

2 E-m [eV]

3

4

FIGURE 72. (a) Comparison of Monte Carlo simulations with Eq. (495) with Ks ¼ 1.684 for copper. (b) Comparison of other theory and measurements with Eq. (495) with Ks ¼ 14.417 for silver. (c) Comparison of Monte Carlo simulations with Eq. (495) with Ks ¼ 6.4546 for gold.

217

ELECTRON EMISSION PHYSICS

Kanter Krolikowski Ks = 14.417 Ks = 2.060

tee [fs]

100

10

1 Ag

0.1

0

2

4

6 E-m [eV]

8

10

FIGURE 73. Same as Figure 72(b) but for a larger E range, showing impact of the d electrons.

(a) 14 Cu

In(relaxation time [fs])

12 10 8 In(t ) In(t ep) In(t ee) In(matt.rule) Liq. nitrogen Room temp

6 4 2 0 −2

0

1

2 3 4 5 In(temperature [kelvin])

6

7

(b) 14 Ag

In(relaxation time [fs])

12 10 8

In(t ) In(t ep) In(t ee) In(matt.rule) Liq. nitrogen Room temp

6 4 2 0

0

1

2 3 4 5 In(temperature [kelvin]) FIGURE 74. (Continues)

6

7

218

KEVIN L. JENSEN

(c) 14 Au

In(relaxation time [fs])

12 10 8 In(t ) In(t ep) In(t ee) In(matt.rule) Liq. nitrogen Room temp

6 4 2 0 −2

0

1

2 3 4 5 In(temperature [kelvin])

6

7

(d)

In(relaxation time [fs])

10

Pb

8 6 4

In(t ) In(t ep)

2

In(t ee) In(TOTL) Liq. nitrogen Room temp

0 −2

0

1

2 3 4 5 In(temperature [kelvin])

6

7

In(relaxation time [fs])

(e) 10 W

8 6 In(t ) In(t ep) In(t ee) In(TOTL) Liq. nitrogen Room temp

4 2 0 −2

0

1

2 3 4 5 In(temperature [kelvin])

6

7

FIGURE 74. (a) Determination of the deformation potential and the defect scattering term from thermal conductivity data from which the relaxation time is taken for copper. Room temperature and liquid nitrogen temperature (300 K and 77 K, respectively) are also shown. (b) Same as (a) but for silver. (c) Same as (a) but for gold. (d) Same as (a) but for lead. (e) Same as (a) but for tungsten.

ELECTRON EMISSION PHYSICS

219

(2004), the findings of Lugovskoy and Bray (1998) for copper, and the theory and measurements of Kanter (1970) and Krolikowski and Spicer (1969) for gold. A first take suggests that the Ks modification makes practical sense given the ability to account for the data, but such an assessment is premature. The first hint of complications is the manner in which Ks changes from metal to metal in the cases considered. As noted separately by Lugovskoy and Bray (1998) and Krolikowski and Spicer (1969) and alluded to previously, complications occur as a consequence of the d electrons that change the effective value of Ks, as can be seen by examining a greater range of energies as for silver (Figure 73). A transition occurs in the scattering rate that can be modeled by a change in the value of Ks as the higher‐energy photons probe more deeply into a DOS that is at variance with the nearly free electron model. Nevertheless, the theoretical description can be adapted to the regime of interest through a modification of Ks. If tdef and Ks can be independently ascertained, then the final specification of X using thermal conductivity data is straightforward. If that option is not viable, then a least‐squares minimization procedure can be used to form adequate estimates from the thermal conductivity data, although such a procedure is less satisfactory because of the domination of te‐e by te‐p for scattering at the Fermi level. The results of such a calculation are shown in Figure 74 for copper, silver, gold, lead, and tungsten, where the open circles represent t as calculated from thermal conductivity data via Eq. (592) and ‘‘Matt.rule’’ represents Eq. (591).

z q

x

Vacuum

Surface

Cathode

−hω FIGURE 75. Relation of the parameters z, x, and y for the path of a photoexcited electron inside a metal.

220

KEVIN L. JENSEN

G. Scattering Factor With an estimate for the relaxation time available, an estimate of the impact of scattering on the QE of a metal can be made. The correct approach should be to consider the scattering rate as a function of electron energy after it absorbs a photon and then sum over the contributions of all electrons that are photoexcited and satisfy conditions for emission, a method to be considered below and designated the moments approach. It is possible, however, to form an estimate of the impact of scattering on QE in the context of the Fowler–Dubridge model if it is assumed that (1) the electron energy prior to photoexcitation is concentrated around the Fermi level, and (2) the emission probability and the transport to the surface are unrelated. Both approximations are not strictly true, but they allow for the decomposition of the QE into the product of an emission probability P and a scattering factor Fl, of which the former was considered previously and the latter the topic of the present inquiry. The presence of scattering introduces a factor that accounts for the probability an electron will migrate to the surface with a kinetic energy component normal to the surface sufficient to be emitted given by a weighted average of the product of the probability that an electron will absorb a photon at a depth x in the material and the probability that the electron will not scatter before reaching the surface. Two modifications are that, first, only those photoexcited electrons with energy sufficient for emission are considered, and second, only those electrons with a velocity component toward the surface are considered (Figure 75). Such a demarcation of the photoemission process into absorption, transport, and emission is familiar from the successful three‐step model of photoemission introduced by Spicer (Spicer, 1960; Sommer and Spicer, 1965; Spicer and Herrera‐Gomez, 1993). Consider the simpler question of the number of electrons, expressed as a fraction of the number photoexcited, that reach the surface without suffering a scattering event under the approximation that any scattering event is fatal to emission; while not strictly correct and perhaps draconian, as electrons can scatter into states that can be emitted in addition to being scattered out of such states, nevertheless when the energy of the electron is sufficiently above the Fermi level, electron‐electron scattering dominates in metals, and collisions with other electrons generally divide the energy. The approximation therefore has more in its favor than against it—the existence of a surface barrier and the restriction of attention to incident photons with energies not much larger than the barrier height make the approximation quite reasonable. Let the incident photon be absorbed at a depth x with a probability proportional to exp(–x/d), where d is the penetration depth. Let the mean distance the electron travels in any direction before suffering a collision be

ELECTRON EMISSION PHYSICS

l ðkÞ ¼


jkj h tðEðkÞÞ; m

221 ð593Þ

where if scattering is isotropic, then only the magnitude of k need be considered. The probability that an electron will suffer a collision after traveling l(k) is then proportional to exp(–z(y)/l(k)). Consequently, the fraction of the electrons reaching the surface is approximately
 ð1 Ð x zðyÞ exp   dx dkfFD ðEðkÞÞYðEðkÞ þ
ho  m  fÞ d l ðkÞ 0 ð1 Fl ¼ :   Ð x exp  dx dkfFD ðEðkÞÞYðEðkÞ þ
ho  m  fÞ d 0 ð594Þ A number of subtle features exist. The first is that because only electrons with an x‐component directed at the barrier are considered, the angular integration is to p/2 instead of p, whereas in the denominator any direction can result after photoexcitation. The second feature entailed by the Heaviside step function is not just any photoexcited electron but only those with an energy that would allow them to surmount the barrier if they were favorably directed, contribute to the QE (at present, the possibility of a tunneling component to the emission is ignored). Recall that the probability of emission factor P containing the Fowler–Dubridge functions has already dispensed with those electrons that cannot pass the surface barrier, and the emphasis here is only on the remaining electrons and what additional processes they endure. With the identification that z ¼ x=cosðyÞ, the x‐integrations in both numerator and denominator are trivial, and result in a multiplicative factor of

 ð1 1 1 exp x þ dx cosðyÞ d lðkÞcosðyÞ 0 ð1 : ð595Þ ¼  x cosðyÞ þ ðd=lðkÞÞ exp  dx d 0 The angular integration is likewise analytic, giving

   ð 1 p=2 cosy 1 d d 1þ sinydy ¼ ln : 2 0 cosy þ ðd=l Þ 2 l dþl

ð596Þ

The final momentum integration calls for an understanding of the momentum dependence of the relaxation time, which is complicated beyond the needs of a simple approximation. Brief reflection indicates that the majority of the photoemitted electrons have energies not much larger than the barrier height, and so a reasonable approximation is to replace k by

222

KEVIN L. JENSEN

10−3

Quantum efficiency

Cu

10−4

QE (theory) Rosenzweig et al. Rao et al. Dowell et al.

10−5

3.9

4.1

4.3

4.5

Work function Φ [eV] FIGURE 76. Quantum efficiency as predicted by the modified Fowler–Dubridge model of Eq. (599) compared to experiment for copper.

kv ¼ ½2mðm þ fÞ1=2 =
h in l(k), thus allowing the final k integration to be the same in the numerator and denominator and factor out. It follows 8 0 19 = 1< d d A 1þ ln@ Fl  2: lðkv Þ d þ lðkv Þ ; ð597Þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2mðm þ fÞ kv ¼ h
A numerical example is profitable. Assume that the dominant part of the relaxation time is due to electron‐electron scattering, and use the low‐ temperature leading‐order limit, for which 
m 2Ks ðm þ fÞ 1=2 h tee ðkv Þ  32 2 : afs h
kF cp3 f

ð598Þ

Using copper parameters and neglecting field, it is found that l is approximately 3.6 nm, whereas d is 12.9 nm. This suggests that Fl ¼ 0.059. With the crude estimation provided by Eqs. (593)–(598), a first‐pass estimate of the QE may be made using the low‐temperature expansion of the Fowler– Dubridge functions QE  ð1  RÞFl



ho  F 2 : m

ð599Þ

ELECTRON EMISSION PHYSICS

223

Again use copper parameters of m ¼ 7.0 eV and R ¼ 34% at l ¼ 266 nm. Work function varies with crystal face, a common value of F ¼ 4.48 eV for the 110 face (Weast, 1988). The predictions of the approximation to QE given in Eq. (599) are shown in Figure 76 for a range of work function values. Compare this to three reported values in the literature (i.e., Table I of Srinivasan‐Rao, Fischer, and Tsang, 1991; Figure 6 of Rosenzweig et al., 1994; and Figure 8 of Dowell et al. 2006). These values are shown on the labeled points at the work function of 4.3 eV (the value taken for the photoelectric work function of copper by Srinivasan‐Rao and Dowell—though the field for Rosenzweig (1994) was large enough to cause a consequential Schottky barrier–lowering factor). As discussed by Dowell et al. (2006), much variation can result as a consequence of surface preparation, so the seemingly dissimilar experimental values are not surprising. What is notable, instead, is how closely the crude estimate Eq. (599) for QE actually approaches measured values. The estimate is improved in the moments‐based approach discussed below. H. Temperature of a Laser‐Illuminated Surface 1. Photocathodes and Drive Lasers The QE of many materials is generally no better than a few percent (although some, such as cesiated GaAs, can be high as 40%), and in the case of metals, the QE is often on the order of 0.001% (Rao et al., 2006). As expected, the application dictates the photocathode, although the applications that motivate the present treatment principally are those that demand high peak and average current densities, such as particle accelerators (Schmerge et al., 2006) and high‐power free‐electron lasers (O’Shea and Freund, 2001; O’Shea et al., 1993). While the QE of the semiconductor photocathodes is without parallel, such photocathodes have response times that are longer than picoseconds and so, if ultrashort bunches (sub‐picosecond or femtosecond; Riffe et al., 1993) and crisply defined laser pulse profiles (sub‐picosecond rise and fall times) are demanded to produce bunches from rugged photocathodes, metal photocathodes are required. Alternately, the generation of polarized electron beams (Maruyama et al., 1989) requires cesiated GaAs photocathodes characteristic of, for example, the JLAB DC photoinjector. RF injectors, owing to their generally more hostile environment, tend to rely on metal photocathodes such as copper (Rosenzweig et al., 1994). The consequences of attempting to extract a charge bunch require use of a laser pulse. As shown in Table 10, such an ability is constrained as much by the drive laser as by the QE. The wavelength of a drive laser is obtained by nonlinear conversion crystals that reduce (for sake of argument) the

224

KEVIN L. JENSEN

wavelength of a 1064‐nm laser by doubled (512 nm), tripled (355 nm), or quadrupled (266 nm) Nd:YAG conversion, with conversion efficiencies of approximately 50%, 30%, or 10%, respectively (Jensen et al., 2003a,b). For the UV case, therefore, a substantial amount of waste heat is dumped into the crystals, altering their operation and leading to nonlinear performance. This effect is generally undesirable as the nonlinear conversion process introduces fluctuations that scale as (laser intensity)n, where n is the harmonic number (4 for 266 nm), thereby causing such fluctuations to appear in the resulting electron pulses. Noise of that character results in degraded FEL operation. Even if the electron bunches are so separated in time that heat generated at the cathode can be dissipated between bunches, as for accelerators, it is still important to question what impact a local, short‐ duration laser pulse will have on the temperature of the electron gas, as the underlying processes, from escape probability to scattering factors, are all dependent on the temperature. In fact, extreme laser intensities for very short durations reveal nonlinear effects of considerable interest and much else (Agranat, Anisimov, and Makshantsev, 1988, 1992; Fujimoto et al., 1984; Girardeau‐Montaut, Girardeau‐Montaut, Moustaizis, and Fotakis, 1994; Girardeau‐Montaut and Girardeau‐Montaut, 1995; Imamura et al., 1968; Papadogiannis and Moustaizis, 2001; Papadogiannis et al., 2002; Yibas and Arif, 2006). 2. A Simple Model of Temperature Increase Due to a Laser Pulse For a crude approximation of the temperature rise from extracting 1 nC from a surface area of (1/8) cm2 under the assumption that all the energy deposited on the surface to extract that amount of charge is uniformly distributed over 2500

Temperature [K]

2000 Electrons Lattice

1500 1000 500 0

0

1

2 3 4 5 Peak intensity [GW/cm2]

6

7

FIGURE 77. Calculated peak temperature for copper subjected to 248‐nm wavelength 450‐ps laser pulse (after Figure 3 of Papadogiannis, Moustaizis, and Girardeau‐Montaut, 1997).

ELECTRON EMISSION PHYSICS

225

a slab of thickness equal to the laser penetration depth 12.93 nm for copper, consider the following model. The temperature rise is related to the specific heat, the amount being heated, and the energy deposited, or DT ¼

DE ; VCi ðTÞ

ð600Þ

where V ¼ Ad is the volume and Ci (T) is defined in Eq. (456) (often, the replacement Ci ¼ ci di, where ci is the specific heat in units of joules per gram Kelvin [J/gK] and di is the mass density, results in the more familiar DT ¼ DE=Mci ðTÞ, where M is the total mass involved). The Debye temperature of copper is larger than room temperature (343 K), and so an approximation to Ci of Ci ðT Þ 

3kB ri
 1 TD 2 1þ 20 T

ð601Þ

is useful. It follows
 "
 #
ho DQ 1 1 TD 2 DT ¼ 1þ ; q AdkB ri QE 20 T

ð602Þ

which, for a wavelength of 266 nm and DQ ¼ 1 nC is approximately 62.4 K. If the laser pulse is of 50 ps in duration, the intensity of the laser is 5 MW/cm2. While 60 K temperature rises indicate some interesting physics is in store, clearly if the laser pulse illuminates a smaller area and is of significantly greater intensity (see, for example, Papadogiannis and Moustaizis, 2001, in which >1 GW/cm2 intensities are used), then the temperature excursion can be significant, and the metal brought to high temperatures that give rise to thermionic emission (Riffe et al., 1993) and show evidence of a decoupling between the temperature of the electron gas and the lattice. Such a state of affairs is shown in Figure 77 (which summarizes Figure 3 of Papadogiannis, Moustaizis, and Girardeau‐Montaut, 1997). Although Eq. (602) is pedagogically appealing, it is clearly incorrect in its assumptions that the laser energy is uniformly distributed over a depth d and that such a depth is independent of the duration of the laser pulse. In what follows, methods to model the temperature rise and the impact on photoemission are considered that allow for the impact of sudden pulses on the rise in temperature near the surface and how that relates to both energy transfer to the lattice and thermal diffusion into the bulk, always with an

226

KEVIN L. JENSEN

eye toward returning the discussion to the treatment of QE and laser heating of a photocathode (Jensen, Feldman, Moody, and O’Shea, 2006a) at the appropriate time. 3. Diffusion of Heat and Corresponding Temperature Rise Thermal current density harkens back to the defining relations in Eq. (435), and like charge current density, it obeys a continuity equation analogous to Eq. (126) that relates the time rate of change of a density to the spatial variation in the current. The ‘‘density’’ is the energy density, the time derivative of which is related to the specific heat, whereas the spatial derivative of the current is ascertained from Eq. (433), a form of Fick’s law relating a ‘‘current’’ to a spatial variation in a density that dominates the treatment of diffusion and transport phenomena: the current across a surface is related to its gradient analogous to Eq. (526). Therefore, a combination of Fick’s law and the continuity equation entails that a region where the thermal energy is concentrated diffuses into regions where it is not, according to
 @ @ @ C v ðT Þ T ¼ kðTÞ T ; ð603Þ @t @x @x where T is the temperature of the electron gas, Cv is the specific heat, and k(T) is the thermal conductivity (e.g., Cv ¼ Ce(T) þ Ci(T) ¼ 3.45 J/Kcm3 and k ¼ 401 W/mK for copper at room temperature). Although Eq. (603) somewhat dominates the discussion, at the outset it is apparent that it cannot be quite right for several reasons. First, the equation needs a source term representing the drive laser. Second, since the electron and lattice temperature can decouple, and the fact that the relaxation time depends on both e‐e and e‐p scattering rates, the thermal conductivity actually should be kðTe ; Ti Þ, where e indicates the electron temperature and i the temperature of the phonon bath (reflecting a slavish obedience to conventions established in the literature). Finally, if the electron and lattice temperature differ, then a term accounting for the bleeding off of electron energy to the lattice as electrons and phonons interact must be included. These complications will appear after the simple form of Eq. (603) is examined. The first and simplest approximation is to assume that the thermal conductivity is at best weakly dependent on temperature so that @x ðk@x T Þ  k@x2 T, and second, that the temperature excursions are small, so that k=Ce  Do is approximately constant, where Do has units of square centimeters per second (cm2/s). Solutions exist of the form Tðx; tÞ ¼ To þ co DTuðx; tÞ @t uðx; tÞ ¼ Do @x2 uðx; tÞ

ð604Þ

ELECTRON EMISSION PHYSICS

227

where DT and To are a temperature rise and the baseline (or bulk) temperature, respectively, and where the parameter co is a constant to act as a placeholder for future factors that will invariably arise but which are inconvenient to specify now. Let w be the spatial Fourier transform of u such that ð 1 1 uðx; tÞeikx dx; ð605Þ wðk; tÞ ¼ pffiffiffiffiffiffi 2p 1 then   @t wðk; tÞ ¼ Do k2 wðk; tÞ ) wðk; tÞ ¼ wo exp Do k2 t :

ð606Þ

Inverting the Fourier transform and normalizing u so that its integral over all space is unity gives

 uðx; tÞ ¼ ð4pDo tÞ1=2 exp x2 =ð4Do tÞ :

ð607Þ

A feature exploited below is that for large Dot, the u function acts remarkably like a Dirac delta function (in point of fact, the derivation of Eq. (607) is a useful approach to ‘‘deriving’’ the properties of the delta function; see, for example, Butkov, 1968). Hence, Eq. (607) shall be referred to as a delta‐function–like pulse, not because it is so sharp but because for small times, when integrated with other x‐dependent functions, it behaves in a way that mimics a delta function, even though it is Gaussian when the time parameter is large. On a related note, the solution entailed by Eq. (604) is analogous to the path integral formalism of quantum mechanics (Rammer, 2004), as the heat diffusion equation and Schro¨dinger’s equation are formally analogous, but where the real temperature in the former is the imaginary time in the latter. From a macroscopic viewpoint in which the laser penetration depth is as good as infinitesimally thin, the dumping of a quantity of energy in an infinitesimally short pulse creates a temperature spike that proceeds to diffuse into the solid. Near the surface, though, a complication arises in that heat does not diffuse from the solid into the vacuum (it radiates—but that is ignored for now), or equivalently, the boundary condition that the gradient of temperature at the surface vanishes is imposed. If the pulse is absorbed some distance xo in the surface, then a method to ensure the boundary condition is to add an image pulse a distance xo outside the surface, so that ( ! !) 1 ðx þ xo Þ2 ðx  xo Þ2 uðx; tÞ ) pffiffiffiffiffiffiffiffiffiffiffiffiffi exp  þ exp  ; ð608Þ 4Do t 4Do t 4pDo t where the gradient at x ¼ 0 vanishes. In the limit that xo approaches

228

KEVIN L. JENSEN

0 (the pulses are absorbed at the surface), then the modification is to insert a factor of 2, one of the many small factors that are absorbed into the definition of co at the appropriate time. 4. Multiple Pulses and Temperature Rise Insofar as a pulse of arbitrary duration may be considered the sum of many infinitesimal pulses, it is relevant to ask how such pulses sum. If our viewpoint is enlarged to even longer times, then the pulse again appears to be like Eq. (607), and perhaps that would suffice, but it glosses over an important feature: rather than being an academic exercise for obscurantist theorists, a train of equivalent pulses is what a photocathode endures in the operation of an FEL or accelerator, and so the question of the cumulative rise in temperature becomes related to the time separation between pulses and the energy content of each pulse, apart from what happens in a particular finite duration pulse, though that is critical as well. Consider a train of Dirac delta‐function–like pulses, where each individual pulse gives rise to a term like u(x,tn) for the nth pulse. The temperature as a function of position and time is then the sum over such pulses, and it matters whether the time of interest is during or after the period when the train of pulses is incident on a surface. It is an initial assumption that the coefficients co and DT are the same for each pulse; that this cannot be strictly true is evident because as the temperature rises, the relaxation rates change and therefore the conductivity changes, but to leading order and especially if the energy content of each pulse is small, the approximation is quite reasonable. Therefore, the temperature can be written Tðx; tÞ  To þ co DT fSn ða ðxÞ; sðtÞÞ  Sn ðaþ ðxÞ; sðtÞÞg 2 3 Xn 1 a 5 exp4 Sn ða; sÞ  j¼1 nþs ð j þ sÞ1=2

ð609Þ

where the difference in S functions arises because the back boundary of the cathode of finite thickness is to be held to the boundary condition that the temperature there is To. New terms a and s have been introduced. They are defined as follows. Time is a function of a characteristic time Dt (the pulse‐to‐pulse separation), a pulse number index n, and an offset parameter s that will be (1/2) for times in between adjacent pulses or odd multiples of (1/2) for times after the last pulse in a pulse train, or t ¼ tn ðsÞ ¼ ðn þ sÞDt. It follows that for a total number of pulses N, if n < N, then s ¼ 1/2 and the time period corresponds to heating due to absorbed laser pulses, but if n  N, then s ¼ ðn  N þ 1=2Þ and the time period is one of cooling after the last pulse has been absorbed and time elapses. Next, let the width of the cathode

ELECTRON EMISSION PHYSICS

229

be L and the position x be a function of a dimensionless term y such that xðyÞ ¼ ð1  yÞL so that y ¼ 0 corresponds to the back contact and y ¼ 1 corresponds to the surface. A fictitious image pulse is needed equidistant from the back contact, corresponding to x ¼ 2L, so that the boundary conditions of holding the back contact at fixed temperature can be maintained [hence the aþ term in Eq. (609)]. Thus, a ðyÞ ¼

ð 1 yÞ 2 L 2  ao ð 1 yÞ 2 : 4Do Dt

ð610Þ

In the limit of large N, converting the summations to integrals shows that 8 2 3 2 39 sffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffi = pffiffiffiffiffiffi< a 5 a5  Erf 4 SN ða; sÞ  2 pa Erf 4 : sþN s ; ð611Þ ð4ðs þ NÞ þ 1Þ a=ðsþNÞ ð4s  1Þ a=s pffiffiffiffiffiffiffiffiffiffiffiffi e pffiffi e þ  2 s 2 sþN where the error function is defined by 2 Erf ðzÞ  pffiffiffi p

ðz 0

  exp x2 dx

ð612Þ

and where the extra terms in Eq. (611) arise from the application of the trapezoidal rule endpoints, which cannot be ignored when converting the summation to an integral. Two cases are of particular interest—first, early in the pulse train or when the pulse train is short; and second, when the pulse train is so long that a disturbance has propagated to the back of the slab. Treating the first case first, using copperlike parameters at room temperature Do ðCuÞ ¼

kðTÞ 4:01W =cmK cm2  ¼ 1:16 : Cv ðTÞ 3:45 J=cm3 K s

ð613Þ

Consequently, a copperlike slab roughly half of a millimeter thick subject to pulses roughly 1 ns in duration entails ao ¼ 500,000—assuredly a big number, but one that pales in comparison to the number of pulses (109) that make up a 1‐second engagement. Early in the train, however, when the number of pulses is small compared to a, meaning the ratio a=ðN þ sÞ is large, and using the approximation to the error function for large argument
 expðx2 Þ 1 pffiffiffi Erf ðx  1Þ  1  1 2 ; ð614Þ 2x x p

230

KEVIN L. JENSEN

kth term of SN(a −) - SN(a +)

(a) 1.5 k k k k k

0.1

= = = = =

0 1 4 8 64

a o = 100 s = 1/2

0.5

0.0 0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

y

SN(a o(1−y)2,s)-SN(a o(1+y)2,s)

(b) 50 N=1 N=4 N = 16 N = 64 N = 256 N = 1024 Equilibrium

40 30

ao = 100 s = 1/2

20 10 0 0

0.2

0.4 y

FIGURE 78. (a) Components of Dirac delta function–like thermal pulses at different times. (b) The sum over the pulses shown in (a) for various total times.

then noting that if a/(Nþs) is large, then a/s is far larger, it can be shown that Eq. (611) is well approximated by )
( 1 a ðN þ sÞ2 SN ða; sÞ  pffiffiffiffiffiffiffiffiffiffiffiffi exp  1þ2 ; ð615Þ N þs a 2 N þs which confirms the intuitive judgment that a train of pulses continues to look like an expanding delta‐function–like pulse governed by a relation that closely resembles Eq. (607), albeit that the coefficient has acquired a few numerical constants; that is, the sum of N pulses has a form that resembles one of its summation terms with n replaced by N. Figure 78a shows an example of such an expanding pulse, although the Figure is equally valid if

ELECTRON EMISSION PHYSICS

231

the time coordinate is scaled by a factor ls and the spatial coordinate byl1=2 s , and where the time axis is begun away from the origin at 0.5 so as to not have the Figure dominated by the sharpness of the pulse for earlier times. The second case for consideration is when so many pulses have occurred that heat is being lost to the back fixed‐temperature boundary and equilibrium ensues. Such a condition defines a maximum temperature parameter. In this case, N is asymptotically large and ao/(Nþs) small. For small y, neglecting s by comparison to N, and to order N1/2, a bit of work shows that the small y limit is 0 1 2 3 2Þ 2 2a y a ð 1 þ y o o Aexp4 5 pffiffiffiffiffiffiffiffiffiffiffi sinh@ SN ½a ; s  SN ½aþ ; s ¼ k¼0 k þ s kþs kþs 0 1 sffiffiffiffiffiffiffi ao A pffiffiffiffiffiffiffi@  4 pao 1  2 y pN XN

ð616Þ

In other words, a linear behavior with respect to x occurs at the back boundary as N becomes large. Equilibrium entails time independence, and so a linear function in x is what is expected from Eq. (604) after a long time. The temperature declines linearly from the hot to the cold boundary, as shown in Figure 78b for the example parameters of ao ¼ 100. Setting y ¼ 1 in Eq. (616) defines a characteristic maximum temperature above background given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (

ffi) rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi p L L ¼ co DT 2 3p ¼ co DT 2L ; Do Dt vF Dt vF t


Tmax

ð617Þ

hkF =m and the odd way of writing the RHS where vF is the Fermi velocity
shows that three length scales are involved: the width of the cathode, the mean free path, and how far an electron at the Fermi level travels during the duration of the pulse. Note, however, that we have been rather cavalier with the parameter Dt: it has been treated as a differential element analogous to dt, but the conclusions do not change if it is treated as the FWHM width of one laser pulse or 1/100 of such a pulse—in fact, it could even be much larger than t and the conclusions drawn by Eqs. (616) and (617) would not change. Moreover, nothing has been said about whether adjacent pulses share a common boundary (merge into a larger pulse) or are separated by a time increment that can be much larger than the pulse length itself, so that questions of heating due to a finite train of short‐duration pulses can be investigated, an advantage of the manner in which the problem was formulated.

232

KEVIN L. JENSEN

The final factor needed to estimate Tmax is an expression for DT. If a total amount of energy DE is deposited on the surface of a material, then ð0

ð0

 dt Cv ðTÞ@t T  Do @x2 T L Dt=2 8 2 39 ð0 ð0 < 1 = ¼ dx dt @t 4 gT 2 þ Ci T 5 ; 2 L Dt=2 :

DE  2

dx

ð618Þ

where the assumption is that the energy deposited on the surface is done so symmetrically in time (e.g., a Gaussian laser pulse), the disappearance of the term containing Do is a consequence of @x T ¼ 0 at the boundaries, and the approximation Ce ðTÞ  gT has been used. The time integration is straightforward, and so 3  2  1 DE  2 dx4 g T  To2 þ Ci ðT  To Þ5 2 L ð0 2 dxðT  To ÞfgTo þ Ci g L ð0 ¼ 2Cv ðTo Þ dxðT  To Þ ð0

2

ð619Þ

L

50 S(N,0,1/2) - S(N,4a o,1/2)

Numerical 4N + 2

40

4 pao 1–2 ao pN

30 20 10 0

ao = 50 1

10

100 N

1000

104

FIGURE 79. Comparison of the S functions with the asymptotic values for the evaluation of surface heating.

233

ELECTRON EMISSION PHYSICS

Using the relation Eq. (604), defining matters such that T(0,0) ¼ To þ DT, and taking L to be so large compared to other length scales that the lower limit can be taken to infinity, it follows DT ¼

DE pffiffiffiffiffiffiffiffiffiffiffi ; Cv ðTo Þ Do Dt

ð620Þ

TABLE 11 COPPERLIKE PARAMETERS Parameter

Value

Units

R Ce(300 K) Ci(300 K) g TD F/q Io k Lo Do l Pulse‐to‐pulse Pulse width (FWHM) t ao

33.7 0.0291 3.297 9.7
105 343 10 1 4.007 1 1.20 266 15 10 16.78 2075400

% J/K cm3 J/K cm3 J/K2cm3 K MV/m MW/cm2 W/K cm2 cm cm2/s nm ns ps fs —

FWHM, full width at half maximum.

TABLE 12 GOLD AND COPPER PARAMETERS Parameter

Units

Copper*

Gold

Sound velocity vs Atomic mass M Chemical potential m Lattice temperature Ti Relaxation time [Eq. (638)] gexp{ g [Eq. (627)]/gexp g [Eq. (637)]/gexp

m/s gram/mole eV Kelvin fs GW/K–cm3 — —

4760 63.546 7 1000 20 60 2.38 0.44

3240 196.9665 5.51 1000 36 40 0.64 6.8

From Wright and Gusev, 1995. { From Fann et al., 1992.

234

KEVIN L. JENSEN

Temperature [K]

380 Heating Cooling To Tmax

340

Cu ∆t = 15 ns d tFWHM = 10 ps

300 −8

−6

−4 −2 log10{t [s]}

0

FIGURE 80. Calculation of the temperature rise during laser pulse heating and cooling rate after the last pulse for copperlike parameters.

which allows Tmax to be identified as Tmax  To ¼

pffiffiffi
 pffiffiffi 2L p DE 2L p Il :  Cv ðTo ÞDo Dt kðTo Þ

ð621Þ

Consider the canonical copper example used in Eq. (613) for a 0.5‐cm thick sample subject to a laser intensity of Il ¼ 100 W/cm2: Tmax – To under these conditions is equal to 44 K. Finally, there is the question of how fast the metal heats and how fast it cools once the pulses stop arriving on the surface. Explicit use of the fact that adjacent pulses can be separated in time can now be made: the sum of a train of pulses separated in time by an increment even larger than the pulse width itself is allowed by the formalism leading to Eq. (616). Two asymptotic conditions are of interest for heating: the initial heating and the approach to equilibrium at the surface (y ¼ 1). As shown in Figure 79 for the ad hoc parameters ao ¼ 50, the behavior of heating (s ¼ 1/2) at the surface follows the asymptotic expressions 8 pffiffiffiffiffiffiffiffiffiffiffiffi 2 N þ0s ðN  ao Þ > > < sffiffiffiffiffiffiffi1 4ao A pffiffiffiffiffiffiffi ð622Þ SN ð0; sÞ  SN ð4ao ; sÞ  ðN  ao Þ 4 pao @1  > > : pN

ELECTRON EMISSION PHYSICS

235

rather well. Easily evaluated models aside, actual parameters are of greater pedagogical value. Consider again the canonical case of copper using a photocathode simulation algorithm (Jensen, Feldman, Moody, and O’Shea, 2006a; Moody et al., 2007), to be discussed in greater detail in the modeling of a single pulse, to model the temperature rise and cooling of an illuminated copper surface. Assume that the individual pulses are Gaussian with a FWHM value of 10 ps (corresponding to a Gaussian time parameter of 6 ps) and that the pulses are separated in time by 15 ns. Assume a QE of 0.0056%. Such values correspond to a peak and average current of 12 A/cm2 and 8 mA/cm2, respectively. Finally, for copper, the relevant values of the various needed parameters are given in Tables 11 and 12. Under such conditions, the heating and cooling profiles are shown in Figure 80. 5. Temperature Rise in a Single Pulse: The Coupled Heat Equations Returning to Eq. (603) which, to accommodate the energy that a laser pulse deposits on the surface, must now be written as noted by Papadogiannis, Moustaizis, and Girardeau‐Montaut (1997) as
 @ @ @ C v ðT Þ T ¼ kðTÞ T þ Gðz; tÞ; ð623Þ @t @x @x where the integral of G(z,t) over all time and space is DE, or the energy dumped into the surface per unit area, and is given by 
x=d  e U½bð
ho  fÞ Gðx; tÞ ¼ ð1  RÞIl ðtÞ ; ð624Þ 1 U½bm d where the reflectivity R is a function of incidence angle and Il is the laser intensity per unit area incident on the photocathode. The overly pessimistic term containing the Fowler–Dubridge U functions nominally accounts for energy loss from direct photoemission (i.e., energy not absorbed and transferred to the lattice from scattering); to leading order it is ½ð
ho  fÞ=m2 , so that for photon energies at or near the barrier height, the term is negligible. This equation is correct, however, only if the electrons and the lattice are in thermal equilibrium, and it is quite possible (and widely done for varied reasons; see Girardeau‐Montaut et al., 1996; Kaganov, Lifshitz, and Tanatarov, 1957; Logothetis and Hartman, 1969; Lugovskoy and Bray, 1998, 1999 1999; Lugovskoy, Usmanov, and Zinoviev, 1994; Mcmillan, 1968; Papadogiannis and Moustaizis, 2001; Papadogiannis, Moustaizis, and Girardeau‐Montaut, 1997; Papadogiannis et al., 1997; Riffe et al., 1993; Rosenzweig et al., 1994; Wright and Gusev, 1995; Zhukov et al., 2006) to make laser pulses of sufficient brevity that the electrons heat to temperatures higher than the lattice without the lattice having time to catch

236

KEVIN L. JENSEN

up. In that case, Eq. (624) becomes not one, but two coupled differential equations for the electron and lattice temperature separately, or 0 1 @ @ @ @ kðTe ; Ti Þ Te A  U ðTe ; Ti Þ þ Gðx; tÞ C e ðT e Þ T e ¼ @t @x @x ð625Þ @ Ci ðTi Þ Ti ¼ U ðTe ; Ti Þ @t where U is the transfer in energy from the electrons to the lattice. To cleanly solve Eq. (625), then U would have to be linear in the difference between the electron and lattice temperatures Te – Ti (where the i subscript nominally denotes ‘‘ions’’) and that is an approximation often made, in which the electron‐phonon coupling constant—called g or some variant—is defined by U ðTe ; Ti Þ  gðTe  Ti Þ:

ð626Þ

One can do better than taking g as a constant. In fact, its determination requires careful attention to competing effects and is important beyond our interest in it here; see Corkum et al. (1988) and Kaganov, Lifshitz, and Tanatarov (1957)—who, in articles often cited and possibly rarely seen— obtained the relation 8 9 p2 2 < 1 1 = U ðTe ; Ti Þ ¼ mvs r  :tep ðTe Þ tep ðTi Þ; 6 0 10 1 ð627Þ 2 2 p @ mvs r A@Te  Ti A  6 tep ðTi Þ Ti where m, vs, and r are the electron mass, sound velocity, and electron number density, respectively. Theoretical estimates of U(Te,Ti) have achieved some sophistication (Girardeau‐Montaut and Girardeau‐Montaut, 1995; Mcmillan, 1968). Still, the preference is to cleave to a simpler model, and therefore a method based on a refinement of the approach developed by Kaganov, Lifshitz, and Tanatarov suffices. 6. The Electron‐Phonon Coupling Factor g: A Simple Model Because photoexcited electrons interact in metals via a fast electron‐electron scattering mechanism, an equilibrium temperature among the electrons is achieved rapidly. Electron collisions with the lattice occur with much less frequency, and so the lattice temperature trails the electron temperature. If the electrons and the lattice are in thermal equilibrium, then the scattering

237

ELECTRON EMISSION PHYSICS

operator @c f ¼

1 ð2pÞ3

ð dk2 S ðk1 ; k2 Þfðn12 þ 1Þf1 ð1  f2 Þ  n12 f2 ð1  f1 Þg; ð628Þ

where the FD f and BE n functions have been defined in Eqs. (482), (569), and the S term originated in Eq. (500) but we shall use Eq. (550) preferentially. The delta function in S, namely, dðE1 þ
ho  E2 Þ, entails that if the electron and lattice temperatures are equal, then the collision term is identically 0. To show this, modify past notation slightly so that f1 ! ðex þ 1Þ1 1 0 f 2 ! ð e x þ 1Þ n12 ! ðey  1Þ1

ð629Þ

where x  be ðm  E1 Þ, x0  be ðm  E2 Þ, and y ¼ bi
ho, and where bs ¼ 1=kB Ts and s designates either e or i. It is readily shown that 0

ðn12 þ 1Þ f1 ð1  f2 Þ  n12 f2 ð1  f1 Þ ¼

ex þyx  1 ; x ðe þ 1Þðex0 þ 1Þðey þ 1Þ

ð630Þ

where the indices on either o or y are superfluous and ignored. The delta function indicates that x0 þ y  x ¼ 0 if be ¼ bi, and so Eq. (630) becomes identically 0 (recall that the x’s have opposite signs than the E’s). The change in the electron distribution that occurs when the electron and lattice temperature become separated is mirrored in the change in the phonon distribution. Consider, then, what occurs when, as a consequence of a temperature change in the lattice so that n ! n þ dn. Eq. (628) becomes ð 1 @c f ) n_ ¼ dk2 S ðk1 ; k2 ÞDnff1 ð1  f2 Þ  f2 ð1  f1 Þg ð2pÞ3 20 1 3 ð631Þ ð  2 1 2p   @ A 4 5 a12 dðE1 þ
ho  E2 Þ DnDf dk2 ¼
h ð2pÞ3 where 0

Df ¼

ex x  1 : x ðe þ 1Þðex0 þ 1Þ

ð632Þ

238

KEVIN L. JENSEN

The term Dn arises from a change in temperature in the BE distribution, and so Dn ¼ 

1 ebi
ho

1



1 ebe
ho

1

¼

eðbe bi Þ
ho  1  1Þð1  ebi
ho Þ

ðebe
ho

ðbe  bi Þ
ho ðebi
ho  1Þð1  ebi
ho Þ

ð633Þ

where the second line is the leading‐order change (the subscripts require particularly careful attention); the approximation is reasonable, as o oD, and so ðbe  bi Þ
hoD  1 for generic parameters. For scattering near the Fermi level (that is, x ¼ 0) and using the delta function in Eq. (631), it follows Df ¼

ðebe
ho  1Þ 2ðebe
ho þ 1Þ

and so, using the definition of a in Eq. (548), 0 1 0 13=2 1 @2pA 2 @2mA jaj 2p n_ ðoÞ ¼ ðm þ
hoÞ1=2 DnDf h2
ð2pÞ3
h 0 13=2 0 12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m þ
ho 2 l2 m2 @2mA @
ho A ð b e  bi Þ ¼ 2 bi
ho þ 1Þð1  ebi
ho Þ 9
v hprM
ð e h s

ð634Þ

ð635Þ

to leading order in be  bi , which explains the flexible attitude toward the e and i subscripts on the b’s. Note that r is the electron number density, so that if it is assumed that one atom donates one electron, the product Mr is the same as the mass density of the crystal. With n_ ðoÞ in hand, then the approximate change in energy per unit time and volume is an integral over the product of the change in the number of phonons at a given frequency with the energy of the phonon at that frequency for all available frequencies, or ð oD 4po2 U ðTe ; Ti Þ ¼ ð2pÞ3 ð636Þ n_ ðoÞ
ho 3 do; vs 0 where o ¼ vs k for phonons. Inserting Eq. (635) into Eq. (636) for the case
o  m then after a bit of algebraic effort, and recalling that kB TD ¼
hoD , h it follows
5
 24=3 prðlmÞ2  m  Ti TD U ðTe ; Ti Þ  ðTe  Ti Þ W 6; ; ð637Þ M hTi
TD Ti

239

ELECTRON EMISSION PHYSICS TABLE 13 LASER HEATING OF TUNGSTEN PARAMETERS Parameter

Units

Simulation

Bechtel*

Wavelength Reflectivity Thermal conductivity at 300 K Density Laser penetration depth Sound velocity Ks Debye temperature Chemical potential Thermal mass ratio Electron specific heat at 300 K Lattice specific heat at 300 K Relaxation time at 300 K G Laser penetration depth

nm % W/K‐cm g/cm3 nm m/s — Kelvin eV — J/K–cm3 J/K–cm3 fs GW/K–cm3 nm

1064 60.3184 1.19715 19.3 22.3654 5174 18.0396 400.020 18.08 1.2036 0.04094 2.39981 1.37942 33832.4 22.3654

1060 60 1.78 19.3 25.0 — — — — — — — — — 25.0

*Bechtel, 1975.

which shows the sought‐for linear dependence on the temperature difference between the electrons and the lattice. In computation, l should be evaluated via its definition in Eq. (543) rather than using the Bohm–Staver value of (1/2). The dependence on the W– function and its (T/TD)5 coefficient is hauntingly familiar and appears very similar to the electron‐phonon relaxation time, albeit that there the function W–(5,x) appears. That is,

 1 24=3 plkB T T 4 TD ¼ W 5; : tep ðT Þ
h TD T

ð638Þ

To leading order for small x, the series expansion solutions of W–(n,x) show that
 W ð6; xÞ 4 4 2 x ;  x 1 ð639Þ W ð5; xÞ 5 189 which, to leading order in TD/T, allows Eq. (637) to be cast as 

12 mv2s r Te  Ti U ðTe ; Ti Þ  l2 ðbi mÞ ; 5 tep ðTi Þ Ti

ð640Þ

a form similar to that found by Kaganov, Lifshitz, and Tanatarov (1957), albeit it differs in having a different temperature dependence in the

240

KEVIN L. JENSEN

coefficients because of different approximations for the electron‐phonon relaxation time, but as a pedagogical exercise, the rederivation of the Kaganov form has accomplished its objective of revealing the underlying behavior of the thermal coupling between the electrons and the lattice. Consider, as examples, gold and copper for the parameters given in Tables 12 and 13, where Eq. (637) [rather than the Procrustean Eq. (640)] is compared to Eq. (627)—the comparisons are pedagogical, given the nature of the model and the wide variety of g (and sound velocity) values in the literature—but the agreement is reasonable enough to conclude, first, that the transfer of energy from the electron gas to the lattice is linear in the temperature difference, and second, that the temperature dependence of the coefficient g that governs the transfer follows the temperature dependence of the electron‐phonon relaxation time as found by Kaganov Lifshitz, and Tanatarov and therefore, the widespread use of the Kaganov form (e.g., see Jensen, Feldman, Moody, and O’Shea, 2006a; Papadogiannis, Moustaizis, and Girardeau‐Montaut, 1997; Yilbas, 2006) has merit. I. Numerical Solution of the Coupled Thermal Equations 1. Nature of the Problem The methods used to solve Eq. (625) are rather sophisticated and, through the use of some simplifying approximations about the length of the laser pulse, the temperature variation of the thermal conductivity, and the temperature of the background lattice, analytical solutions are possible, although to make use of them, numerical means are needed to evaluate the terms of the series (Smith, Hostetler, and Norris, 1999). Our goals here are to explore the regime in which the lattice and electron temperatures can diverge, and so numerical methods are sought. A review of methodology is helpful. Solving Eq. (625) using fashionable finite difference methods is a bit premature because a simple finite difference numerical scheme (Smith, 1985) to solve the heat equation @t u ¼ Do @x2 u ) @y u ¼ @z2 u sets limits on the discretization spacing Dy tolerated in the time domain given a discretization spacing Dz in the position domain, where Dy and Dz are normalized variables such that 0 zj 1 with a similar equation for yk, with k and j being index coordinates. Stable and convergent solutions to these parabolic equations for explicit schemes (i.e., ones where the j þ 1 time step is straightforwardly calculated from the j time step solution) are only possible if r

Dy 1 Dx2 , Dt 

; Dz2 2 Do

ð641Þ

241

ELECTRON EMISSION PHYSICS

where the RHS is the largest Dt that can be considered. Taking as example parameters Dx  d=80, where d is the laser penetration depth (on the order of 12 nm) and Do ¼ 1.2 cm2/s, then the largest tolerable time increment is on the order of 0.1 fs. A simulation spanning 50 ps for a 3‐mm thick simulation region would imply Nt ¼ 500,000 and Nx ¼ 20,000, or NtNx2 ¼ 2
1014, and such investments of computer processing time are impractical (apart from the fact that 50 ps is a short time and 3 mm is irrelevantly thin)— even if Do ¼ k=Cv was more or less constant rather than dependent on the evaluation of temperature‐dependent electron‐electron and electron‐phonon relaxation times. In that case, techniques analogous to the multipulse treatment can be brought to bear and much accomplished via analytical means (an excellent example being the analysis of Bechtel, 1975). That, however, is not the situation here, and something more inventive is required. 2. Explicit and Implicit Solutions of Ordinary Differential Equations The resources available that cover numerical issues in the computational solution of partial differential equation cousins, of which the heat equation is a well‐examined representative, are legion (Anderson, Tannehill, and Pletcher, 1984; Smith, 1985). The present interest is in solving such equations when extraordinarily dissimilar time scales and conditions are involved. Over sufficiently small scales, most functions are well approximated by polynomials. The value of functions at various regularly spaced intervals provides a useful estimate of the coefficients of those polynomials, and consequently derivatives of those polynomials, albeit with a greater loss of accuracy the higher the derivative. Let a polynomial P(x) take on the values yj at the discrete points xj  jDx, that is, yj  Pð jDxÞ: Introduce the notation   fn xj  Dxn




d dx

n

  PðxÞ

ð642Þ

:

ð643Þ

x¼xj

A Taylor expansion of P(xj) about x ¼ 0 can then be written XN j n f ð0Þ; yj ¼ n¼1 n! n

ð644Þ

where N is the order of the expansion. A little thought shows that Eq. (644) can be elegantly expressed as a matrix equation when a multitude of yj’s are available, such that the number of columns corresponds to N and the number of rows to the quantity of yj’s available. When the number of rows and columns

242

KEVIN L. JENSEN

are the same, there are N equations for N unknown coefficients of the f ’s so that numerical estimates of the higher‐order derivatives can be made. Define ^ f y ¼M Mjk ¼

jk k!

ð645Þ

For example, choosing points symmetrically about j ¼ 0, then for N ¼ 5 1 0 1 1 0 0 f0 24 48 48 32 16 y2 C B f1 C B 24 24 12 B y1 C 4 1 C B C C B B B y0 C ¼ 1 B 24 B C ð646Þ 0 0 0 0C C  B f2 C C 24 B B @ 24 @ y1 A 24 12 4 1 A @ f3 A 24 48 48 32 16 y2 f4 where, for notational simplicity, the ‘‘(0)’’ has been omitted from the f ’s, and a common divisor has been extracted from the matrix. An obvious symmetry about the center row is evident. The inverse of Eq. (646) gives the 5‐point finite difference approximation, or 1 0 1 0 0 1 f0 y2 0 0 12 0 0 C B 1 B f1 C B 8 0 8 1 C C B y1 C B B C C B f2 C ¼ 1 B 1 B ð647Þ 16 30 16 1 C  B y0 C C B C 24 B @ 6 @ f3 A 12 0 12 6 A @ y1 A 12 48 72 48 12 f4 y2 There is nothing special about equispaced points. Values of the polynomial on the half‐index (i.e., corresponding to xn þ 1/2) or nonuniformly spaced xj are equally subject to the same formalism, although slightly more cleverness is involved. What is less evident, but of greater importance, is that the matrices and vectors of Eq. (646) can be pared to obtain second‐order (i.e., three‐point) approximations by crossing out the nth row and column to eliminate yn to obtain convenient approximations that are useful. For example, the much‐vaunted central difference scheme (CDS) is obtained by eliminating the first and fifth rows and columns, and solving 1 0 0 1 0 1 0 1 0 1 y0 2 2 1 f0 f0 y1 C B 1 @ y0 A ¼ 1 @ 2 ðy1  y1 Þ C ð648Þ 0 0 A  @ f1 A ) @ f1 A ¼ B A @ 2 2 2 2 1 y1 f2 f2 y1  2y0 þ y1 Often, at a boundary (that is, for j ¼ 1 or N), the forward ( j > N) or backward ( j < 1) values are not available (this occurs, for example, if the boundary is absorbing; Jensen and Ganguly, 1993). In this case, upwind and downwind difference schemes are available. Consider explicitly the second‐order upwind difference scheme (SUDS) that follows from eliminating

243

ELECTRON EMISSION PHYSICS

0.64 u(x) DDS SDDS & CDS

u(x)

0.48

u(x) = 1− e−x

0.32

xu 2 xu

= e−x = −e−x

0.16 N=5

0 0

0.2

0.4

0.6

0.8

1

X FIGURE 81. Comparison of the central differencing scheme with a first‐order differencing scheme used at the boundaries downward difference scheme (DDS) with a scheme using second‐ order upwind and downwind differencing schemes (SDDS and central differences scheme (CDS).

the first and second rows and columns and solving for 0 1 2 y0 1 @ y1 A ¼ @ 2 2 2 y2 0

0 2 4

1 1 0 1 0 1 0 y0 0 f0 f0 B1 C 1 A  @ f1 A ) @ f1 A ¼ @ ð3y0 þ 4y1  y2 Þ A: 2 4 f2 f2 y  2y þ y 2

1

0

ð649Þ Observe that to second order, the approximation to f2 has the same structure of subtracting twice the central point from the sum of the endpoints—an indication that a second‐order polynomial has a constant second derivative. The second‐order downwind scheme is trivially obtained by changing the sign of the indices and the second row of Eq. (649). Consider now the usage of finite differencing schemes to solve ordinary differential equations. As a trivial case, consider how to solve the equation @x uðxÞ ¼ vðxÞ with the boundary conditions of u0 and uNþ1. Using the CDS scheme to approximate the first derivative, the matrix version of the equation is 80 1 0 u 1 0 19 0 v 1 > u0 > 1 1 0 1 0 0 0 > > > B . C > . > .. C B .. C B .. C> .. .. .. > > B . . > . C . C . C B . . . = B . C> C B B . C 1 B . 2Dx > . . . . > > C B . . . . . . > @ > > @ .. A . A @ .. A @ .. A> . . . > > > > . ; : 0 0 0 2 2 0 uN vN where only the 1st, jth, and Nth rows are shown. In concise notation,

244

KEVIN L. JENSEN

 1 ^ M u þ ubc ¼ v; 2Dx



ð651Þ

where Mj,j 1 ¼ 1 (except for the last row) and Dx ¼ ðN þ 1Þ1 . The second vector ubc on the LHS is the vector of boundary conditions. Two important but subtle features are noteworthy. First, because it is a first‐order differential equation, Eq. (650) uses only one boundary (u0), which means the other (uN þ 1) must not be included (or vice versa). If the matrix equation is set up so that both of these boundaries are specified, by which the Nth row of the coefficient matrix of u uses the second‐order scheme that the j th row uses, then it is quickly discovered that the coefficient matrix does not have an inverse and a solution is not possible. The second feature is that the solution of the matrix equation is only as good as the worst differencing scheme used. In Eq. (650), the simple, or ^ and so the Euler, downwind difference scheme is used for the Nth row of M, accuracy of the solution is to order Dx, even though the accuracy of the CDS formula is to order Dx2. This is shown in Figure 81 for vðxÞ ¼ ex and uðxÞ ¼ 1  ex and N ¼ 5, for which the first‐order downward difference scheme (DDS) used in the last row of the difference operator matrix in Eq. (650) is responsible for the jagged appearance. If the last row is instead replaced with the second‐order DDS of Eq. (649), that is, instead of ð 0 0 0 2 2 Þ the ^ resembles ð 0 0 1 4 3 Þ, then the second‐order downwind Nth row of M difference scheme (SDDS) line in Figure 81, accurate to Dx2, results. It is worth emphasizing that the order of the solution is dictated by the order, in this case, of the Nth‐row coefficients. If, instead, the equation @x2 uðxÞ ¼ vðxÞ (e.g., Poisson’s equation) were being solved, the matrix version is 80 > 2 1 > > > .. > B .. > . . 1 .. > > >@ .. . > > : 0 0

0 .. .

2 .. . 0

1 0u 1 0 19 0 v 1 > u0 1 1 0 > > B .. C .. C B .. C B .. C> > > C B B . C B . C B . C= B . C C C C B uj C vj C þB 0 C ¼B 1 ... C  B C C: ð652Þ B B > C C B .. A B . C @ .. A> .. B .. C > . > @ . A @ . A . . . > > 1 2 uNþ1 ; uN vN

0 .. .

Note the following: the coefficient matrix is tridiagonal throughout; the boundary vector contains two (not one) boundaries. In the numerical literature, solutions of tridiagonal matrix equations hold a special place, and algorithms to rapidly solve them using a minimum of storage space are widespread and are common in LAPACK1 or IMSL2 software. An example 1 2

http://www.netlib.org/lapack/ http://www.absoft.com/Products/Libraries/imsl.html

245

ELECTRON EMISSION PHYSICS

1 v(x) −u(x)/2 v(x) and u(x)

0.5

0 −0.5 v(x) = sin(2p x) −1 0

5

10

15

20

25

j FIGURE 82. Numerical solution of v(x) compared to its exact representation using the low‐memory solution.

of the solution of the CDS equation of Eq. (652) for the same u(x) considered previously results in the CDS line of Figure 81, which is indistinguishable from the SSDS line. A particularly expeditious solution to @x2 uðxÞ ¼ vðxÞ is possible (Jensen and Buot, 1991) without even numerically defining (i.e., creating) the matrix M. If, on input, the boundary conditions vector is added to the vector Dx2 v  ubc ) v, then the solution in a programming‐like notation becomes For j ¼ 2 to N  j  2vj þ vj1 vj ( jþ1 Next j For j ¼ ðN  1Þ to 1 j vjþ1 vj ( vj þ jþ1 Next j u ¼ v=2

ð653Þ

Figure 82 contains an example for v(x) ¼ sin(2px) and N ¼ 24, where v(x) compares very well to the numerical solution of –u(x). ^ u as considered above. The final step in preparing for the Let @x2 uðxÞ ) M heat equation is to consider solutions to





^ uðtÞ: @t uðtÞ ¼ M

ð654Þ

246

KEVIN L. JENSEN

Symbolically, the solution to this equation is

 ^ uðtÞ; uðt þ dtÞ ¼ exp dtM



ð655Þ

where, as familiar in quantum mechanics (e.g., Eq. (130), and as expected given the formal similarity between the heat equation and Schro¨dinger’s equation), the exponential operator is understood to be replaced by its power series expansion

 X1 dtn  n ^  ^ : exp dtM M n¼1 n!

ð656Þ

That Eq. (654) in Eq. (655) solves Eq. (654) can be verified by substitution. To order dt2, Eq. (656) can be approximated by the Cayley representation (Press, 1992) because  

 ^ 2 þ O dt3 ^ ¼ 1 þ dtM ^ þ 1 dt2M exp dtM 2 0 11 0 1 dt dt ^ A @1 þ M ^A ¼ @1  M 2 2 which implies, with Eq. (655), that

 dt ^ 1 dt ^ uðt þ dtÞ ¼ 1  M 1 þ M uðtÞ: 2 2



ð657Þ

ð658Þ

An alternate method to reach the same result is to use the simple Euler scheme for the time derivative, but using an implicit scheme for the spatial derivative term, where the average of the future and past solutions are joined. Such a scheme is sometimes referred to as the Crank–Nicolson method. Why the average? The spatial derivative should (one would think) be evaluated at the midpoint between the future u(x,tþdt) and past u(x,t) solutions, that is, u(x,tþdt/2), but the midpoint need not be available but is presumably near the average. ‘‘Implicit’’ here is taken to mean that the future value of u is acted on by a nontrivial matrix rather than the identity matrix, and so a matrix inversion is required to solve for u (whereas an ‘‘explicit’’ scheme would have only the identity matrix acting on the future value and therefore require no such inversion). That is, 1 fuðx; t þ dtÞ  uðx; tÞg dt 1 ^ @x2 uðx; tÞ ) M fuðx; t þ dtÞ þ uðx; tÞg 2

@t uðx; tÞ )



ð659Þ

ELECTRON EMISSION PHYSICS

247

which results in Eq. (658) after rearrangement. Implicit schemes can use much larger time steps and still maintain stability, so the added cost of inverting a matrix is often well worth the effort—particularly if an exponentially decaying solution is sought, as exponentially growing solutions frequently satisfy the same differential equation and are otherwise difficult to suppress. If, as shall occur in the heat diffusion case, a source term v(x,t) is added, then, like the spatial derivative term, the average of its future and past values is used rather than simply its past value. A final and widely‐used methodology is based on the ‘‘predictor‐ corrector’’ methods such as that of Runge and Kutta (Press, 1992). A simple Euler scheme might have us conclude, if trying to solve an equation such as @t u ¼ f ðuÞ, where the RHS is a function of the function we are trying to find, that a solution would be

uðt þ dtÞ  uðtÞ þ dtf ðuðtÞÞ ) unþ1 ¼ un þ dtf ðun Þ

ð660Þ

where the second line defers to a simpler notation in which the index refers to the time step. This scheme is accurate only to order O(dt), and that is generally inadequate. A better approach is to take a ‘‘guess’’ as to what unþ1/2 would be and use that in the evaluation of unþ1, or

dt f ðun Þ ) 2   unþ1 ¼ un þ dtf unþ1=2

unþ1=2  un þ

ð661Þ

The accuracy of this approach is substantially better, but one need not stop there, and use a guess to get to the one‐quarter point, use that to guess the half point, and so on, leading up to the ‘‘fourth‐order’’ Runge–Kutta method, which is quite reliable. Thus, of the numerical methods in which we are interested, second‐order differencing schemes for spatial derivatives (CDS away from the boundaries but SDDS and SUDS at the boundaries), coupled with some combination of implicit and predictor‐corrector schemes, may be what is required to avoid the limitations otherwise obstructing our ability to circumvent time scales of widely different magnitude in the solution of the laser‐heated surface. That, as determined below, is in fact a useful approach.

248

KEVIN L. JENSEN

3. Numerically Solving the Coupled Temperature Equations With Temperature‐Dependent Coefficients In the parlance of the previous section, we shall solve for the electron and lattice temperature using both implicit and predictor‐corrector schemes (Jensen, Feldman, Moody, and O’Shea, 2006a). The discrete temporal and spatial coordinates are defined by tj ¼ ð j  1ÞDt and xj ¼ ð j  1ÞDx for 1 j Nt or Nx, respectively, where surface lies at x ¼ 0, and negative x corresponds to the region of space occupied by the photocathode material. For accuracy, the coefficients are temporally averaged as well. The transition from continuum to discrete for a coefficient C(t) and a parameter T(t) proceeds according to (where dependence on x is hidden) CðtÞ@t TðtÞ )

1 ½C ðt þ DtÞ þ C ðtÞ½T ðt þ DtÞ  T ðtÞ: 2Dt

ð662Þ

For the spatial derivatives, the dependence of k on temperature results in @x ½kðxÞ@x TðxÞ ¼

     1  kjþ1 þ kj Tjþ1  Tj  kj þ kj1 Tj  Tj1 : 2Dx2 ð663Þ

If k were constant, the CDS approximation to the second derivative results. The temperatures T are represented as vectors whose j th component corresponds to the spatial coordinate xj; similarly, the coefficients become matrices defined by (where g is the factor from U(Te,Ti) and is approximately constant for high(er) temperature as shown previously; alternately, see Papadogiannis, Moustaizis, and Girardeau‐Montaut, 1997) 1 fCe ½Te  þ Ce ½Te gdlj 2Dt 1 ½Ci l;j ¼ fCi ½Ti  þ Ci ½Ti gdlj 2Dt ½Ce l;j ¼

1 ½Hl;j ¼ gdlj 2 ½Jl;j ¼

g½Ci l;j 2½Ci l;j þ g

ð664Þ

dlj

where dlj is the Kronecker delta function and the temperatures in k are evaluated at the x location at a particular time t. Define

249

ELECTRON EMISSION PHYSICS

Electron temperature [K]

(a) 40 W @1 MW/cm2 30 ns (FWHM) 30

20

10

Te(t)-Tbulk Laser (scaled) Bechtel (fig. 5)

0

−40

−20

0 20 Time [ns]

40

60

Electron temperature [K]

(b) 800 W @ 1 GW/cm2 30 ps (FWHM) 600

400

Te(t)-Tbulk

200

Laser (scaled) Bechtel (fig. 7) 0

−40

−20

0 20 Time [ps]

40

60

FIGURE 83. (a) Calculation of temperature rise for illuminated tungsten surfaces showing the impact of a temperature‐dependent thermal conductivity term (‘‘laser’’) compared to a constant thermal conductivity as done by Bechtel (1975). (b) Same as (a) but for a higher laser intensity over a shorter time.

½DðtÞl;j ¼

     1  ½ k þ k  2 k þ 2k þ k þ k þ k d d dl;j1  jþ1 j l;jþ1 jþ1 j j1 l;j j j1 4Dx2 ð665Þ

250

KEVIN L. JENSEN

Temperature [K]

(a)

700 600 500 400 300 200 100 60 40 20 e [p

Tim

0

s]

−20 −40

−0.25

−0.20

−0.15

−0.05

]

on

icr

m e[

nc

sta

Di

−0.10

(b)

Temperature [K]

2.0 1.5 1.0 0.5 0.0 −0.5 60 40 Tim

20 e [p s]

0

−20

−40

−0.05 −0.10 ] −0.15 on icr m [ −0.20 nce −0.25 sta Di

FIGURE 84. (Continues)

251

ELECTRON EMISSION PHYSICS

Temperature [K]

(c)

Cu: Electrons

800 600 400 200

5

−0.05 0 e [p s]

Tim

−5

Temperature [K]

(d)

−0.25

−0.20

−0.15

n]

cro

i e [m

c

tan

Dis

−0.10

Cu: Lattice

800 600 400 200

5

−0.05 Tim 0 e [p s]

−0.20 −5

−0.25

FIGURE 84. (Continues)

−0.15

−0.10 nc

sta

Di

]

on

icr

e [m

252

KEVIN L. JENSEN

Temperature [K]

(e)

Cu: Difference

800 600 400 200 0 5 Tim

0 s]

e [p

−5

−0.05 −0.10 ] on −0.15 icr [m e −0.20 nc sta −0.25 Di

FIGURE 84. (a) Temperature profile into bulk tungsten for electrons for Bechtel‐like conditions as a function of distance from the surface and time compared to the center of the Gaussian laser pulse. (b) Same as (a) but for difference between electron and lattice temperature in bulk for tungsten. (c) Laser heating of copper for Papadogiannis conditions: electron temperature. (d) Same as (c) but for lattice temperature using same scale; note the differences in peak temperature. (e) Difference between (c) and (d). Note the much greater temperature differences.

as

The matrix form of the coupled temperature equations is then represented

ðCe þ J  DÞjtþDt  Te ðt þ DtÞ ¼ ðCe ð J þ DÞjt  Te ðtÞ þ 2J  Te ðtÞ 1 GðtÞdt þ Tbc þ 2

ð666Þ

ðD þ H ÞjtþDt  Ti ðt þ DtÞ ¼ ðD þ H Þjt  Ti ðtÞ þ H  ðTe ðt þ DtÞ þ Te ðtÞÞ

Ð where Gdt is the integral of the laser term over the time increment, and Tbc accounts for the boundary conditions; far into the bulk, the temperature is held fixed, and at the surface, the gradient of the temperature vanishes. A complex wrinkle to the ‘‘implicit’’ nature of the problem is now evident because the coefficients on the LHS of Eq. (666) must be evaluated at the future time t þ Dt, whereas the temperatures at that time are being solved for and therefore a priori unknown. This is handled by approximating Te and Ti by their values at time t (the guess), solving Eq. (666), and using the

ELECTRON EMISSION PHYSICS

253

predicted values of the temperatures at time t þ Dt to create a new guess to the coefficients (the refinement). The process is iterated several times, the number of iterations being determined by when subsequent refinements have negligible effect. By this ruse, it is possible to choose time steps that are much larger than those tolerated by criteria such as Eq. (641). The numerical solution is decidedly nontrivial to implement, as all manner of terms are dependent on the temperature and particulars of the material parameters that Figure in the scattering terms evaluated at the Fermi level and other quantities discussed throughout this section. The temperature dependence of the thermal conductivity (and other quantities) results in differences in the temperature evolution as compared to solutions where such terms are held fixed, as done by Bechtel (1975). Bechtel considered short laser pulses incident on tungsten for laser intensities of 1 MW/cm2 and 1 GW/cm2 for pulse widths 30 ns and 30 ps in his Figures 5 and 7, respectively. Figure 83 shows the comparison of the numerical solution of Eq. (666) with Bechtel’s findings for the temperature at the surface for the parameters shown in Table 13. To be sure, Bechtel used quantities from the literature (e.g., the AIP Handbook; Gray, 1972), whereas here the same quantities (e.g., relaxation times, specific heat and thermal conductivities, reflectivity, penetration depth) are calculated from the underlying models developed in preceding sections; a comparison of parameters is given in Table 13. Initially, the solutions track reasonably well, but as time progresses, the impact of temperature‐dependent terms becomes evident. The important feature of either Bechtel’s results or the present simulation is that compared to the laser pulse, the temperature maximum occurs after the laser pulse maximum, and the temperature profile is asymmetrical in contrast to the symmetrical laser profile, modeled as a Gaussian with a center at t ¼ 0, as the heat dissipates into the bulk material. Next, consider how heat propagates into the bulk material. The numerical solution of the electron temperature profile for the parameters considered in the 1 GW/cm2 case above results in a temperature profile for the electrons given in Figure 84a, where only a subregion of the entire simulation near the surface is shown. For such parameters, the lattice temperature tracks the electron temperature closely, a consequence of the rapidity of the scattering rates in comparison to the duration of the laser pulse. The difference between the electron temperature and the lattice temperature in such a case is more instructive (as shown in Figure 84b). Here, even though the difference in temperature is never more than a few degrees, the electrons heat up in comparison to the lattice as the laser pulse rises, but after the pulse begins to fade, the electron thermalization causes the temperature to drop below the lattice, at which point the lattice transferring energy back to the electrons prevents their rapid decline in temperature.

254

KEVIN L. JENSEN

Effects are perhaps more evident in the extreme, so consider conditions reminiscent of the high‐intensity studies of Papadogiannis and Moustaizis (2001), in which various metals were subject to GW/cm2‐intensity lasers for very short durations. In such cases, the decoupling between the lattice temperature and the electron temperature is far more pronounced. Here, a Gaussian laser pulse of the form Il ðtÞ ¼ Io exp½ðt=dtÞ2  with dt ¼ 2 ps and an intensity of 3 GW/cm2 under a field of 1 MV/m is incident on a bare copper surface for a wavelength of 266 nm (the reflectivity of copper in IR is quite high, so that copper photocathodes are generally subject to the fourth harmonic of an Nd:YAG laser for which the wavelength is 1064 nm/n, where n is the harmonic number). Now, and in support of similar findings by Papadogiannis et al. (2001), the temperature rise of the electron gas is rather substantial—on the order of 1000 K. However, unlike the case for tungsten where the pulse was both longer and far weaker, now the heating of the lattice follows the electron gas with a lag so that it remains hotter than the electrons past the pulse even though it does not experience nearly as large a temperature rise. Consequently, the temperature of the electron gas is kept high by the lattice returning energy to the electrons after the laser pulse is over. The difference between the electron and lattice temperature in Figures 84c and d, respectively, is shown in Figure 84e. Under such circumstances, metals can be raised to a high enough temperature that thermionic emission can result and complicate the interpretation of whether the electron emission is photoemission or thermionic emission in nature (Bechtel, Smith, and Bloembergen, 1977)—or, for that matter, when coupled with very high fields, to what extent field emission contributes (Brau, 1997; Jensen, Feldman, and O’Shea, 2005). Determining which is which shall be taken up below. J. Revisions to the Modified Fowler–Dubridge Model: Quantum Effects The methodology introduced in the general thermal field equation can be extended to the modified Fowler–Dubridge model to assess the impact of the transmission probability not being a step function on the QE. With the development of the moments‐based approach in the next section, such treatment is perhaps ancillary but is given for aesthetic completeness. For photoemission, the N function introduced in the GTF equation is ðu ln½1 þ enðxþsÞ  N ðn; s; uÞ  n dx; ð667Þ 1 þ ex 1 where u ¼ bF ðEm Þðm þ f 
hoÞ, s ¼ bF ðEm Þð
ho  fÞ. In particular, and in contrast to the GTF equation, Em ¼ m þ f under all conditions as the emission is dominated by electrons passing over the barrier. Therefore,

ELECTRON EMISSION PHYSICS

255

since nothing is added by retaining the argument of bF in the case of photoemission, it is neglected here so that bF without an argument refers to the quadratic approximation, that is, bF ¼ bF ðm þ fÞ. As before, N separates into regions: N ðn; s; uÞ ¼ N1 ðn; s; uÞ  N2 ðn; s; uÞ þ N3 ðn; s; uÞ þ N4 ðn; s; uÞ; ð668Þ where the sign on N2 deserves note. Observe that these are not the same integrals obtained by simply changing the sign of s; rather, they are regions defined according to whether a closed‐form series representation of the integrand components is allowed. N1 and N2 can be done exactly 8 s 9 <ð =

N1 ðn; s; uÞ ¼ n ln 1 þ enðzþsÞ dz ¼ U ð0Þ : ; 1 8u 9 ð669Þ < ð ln½1 þ enðzþsÞ  = dz N2 ðn; s; uÞ ¼ n : ; ez þ 1 x

¼ n2 fU ðsÞ  U ðuÞ  ðu þ sÞ ln½1 þ eu g  n2 U ðsÞ N3 is given by (the perfunctory treatment being a direct consequence of having explained the series methodology in the context of the GTF equation) 8u 9 0 1 < ð nð z þ s Þ = X 1 k N3 ðn; s; uÞ ¼ n dz  ð1Þkþ1 eks Z @ A z k¼1 : e þ1 ; n s ð670Þ jþ1 1 X ð1Þ Z ðxÞ  j ð j þ xÞ j¼1 The fourth term N4 is difficult but can be shown to be 8u 9 < ð ln½1 þ enðzþsÞ  = dz ð671Þ N4 ðn; s; uÞ ¼ n : ; ez þ 1 s 8 9 0 1 < = kns X

1 e 1 k ks @ kA 2 ð 1 Þ e Z  1 þ kn ð kZ ð kn Þ þ Z ð kn Þ Þ þ  zð2Þ þ k¼1 : ; k2 2 n where throughout, by virtue of the largeness of u, terms such as e–u have been neglected. For large s, only the k ¼ 1 terms and terms of order e–s at most need consideration. Combining and keeping dominant terms shows that the photoemission extension to the N function is

256

KEVIN L. JENSEN

Ratio with ref. ( o) value

100

Field = 100 MV/m; lo = 200 nm FD (Cu) GP (Cu) FD (CsCu) GP(CsCu)

10−1

T = 600 K Φ = 1.8 eV

10−2 T = 300 K Φ = 4.5 eV

10−3 200

400 600 Wavelength [nm]

800

FIGURE 85. Comparison of the modified Fowler–Dubridge formulation (FD) with the general photoemission formulation of Eq. (674)


 1 1 N ðn; s; uÞ  n2 s2 þ zð2Þ½n2 þ 1  es n2 S ð672Þ þ ens SðnÞ ; 2 n where for standard technologically accessible photoemission conditions, n is in general smaller than unity. Consider as an example, for cesium on copper (m ¼ 7 eV, F ¼ 1.8 eV) for room‐temperature conditions and a field of 100 MV/m (n ¼ 0.552, f ¼ 1.42 eV) and a photon wavelength of 800 nm (
ho¼1.55 eV). The three separate groupings in Eq. (672) are then 12.5, 2.15, and 0.0114, respectively, the sum of which is 14.6. The MFD equation does not consider currents per se as does the GTF equation, but rather probabilities, and it is therefore the ratio Pð
hoÞ ¼ J ðF ; T; h
oÞ=Jmax ð
hoÞ that is of interest, where the numerator is obtained from Eq. (672) by appending the requisite coefficients on the N function as done for the thermal‐field equation. Following the arguments familiar from the evaluation of the scattering rates, an electron will not be photoexcited unless its final state above the Fermi level is unoccupied, after which only those electrons with a momentum component toward the surface are of importance. Therefore ð ð p=2 ARLD 1 Jmax ¼ 2 2 EfFD ðE Þ½1  fFD ðE þ
hoÞdE sinydy kB 0 0 ð673Þ ARLD  2
hoð2m 
ho Þ kB

ELECTRON EMISSION PHYSICS

257

The general photoemission equation that updates the MFD equation, as the GTF did with the FN and RLD equations, is obtained by replacing the probability of photoemission in expressions for QE with 2 U ðbT ð
ho  f Þ Þ ð
ho  fÞ2 þ 2b2 T zð2Þð1 þ n Þ : ) Pð
hoÞ ¼ U ðbT mÞ 2
hoð2m 
hoÞ

ð674Þ

If the LHS of Eq. (674) is designated as PFD ð
hoÞ and the RHF PGP ð
hoÞ (where FD and GP indicate Fowler—Dubridge and general photoemission, respectively), then a measure of the impact of Eq. (674) is obtained by considering PFD ð
hoÞ=PFD ð
hoo Þ compared toPGP ð
hoÞ=PGP ð
hoo Þ, where oo is an ad hoc reference frequency (here chosen to correspond to a wavelength of 200 nm); the results are shown in Figure 85. For metallic‐like parameters, the Fowler–Dubridge model is adequate; for a low work function surface, however, the quantum effects distinctively make their impact known. K. Quantum Efficiency Revisited: A Moments‐Based Approach The elements of the three‐step model introduced in the discussions of Eq. (376) and combined in Eq. (599) are now reconsidered. As before, the problem of photoemission naturally partitions itself into absorption, transport, and escape. Recall that the modified Fowler–Dubridge approach to estimate QE focused on an independent estimation of each from which QE was obtained by considering their product. Reflection suggests that treating such processes as distinct may be lacking for several reasons. First, treating the transmission probability (leading to the Fowler–Dubridge functions) as a strictly 1D model is at odds with the consideration of the scattering factor Fl as a 3D construct. Second, it would be superior to account for the emission probability in light of the fact that only part of its total momentum is directed at the surface, rather than all. Finally, the formulation of the modified Fowler–Dubridge formula, for reasons intimately connected with these issues, is not adaptable to the evaluation of beam emittance at the cathode in the discussion of thermal emittance; the transverse momentum components are ignored when they are of central importance. Recall that in the development of the 1D current‐density equations, the concept of a supply function was invoked, in which the integrations over the momentum components parallel to the plane of the surface were performed on the FD function characterizing the electron distribution. This was a consequence of the transmission probability being dependent only on the momentum normal to the surface. With photoemission—as noted in the derivation of the Fowler–Dubridge model—the electron energy is augmented by the photon energy. However, the approximation in Fowler–Dubridge that the entire energy of the photon was manifested in the forward direction of the

258

KEVIN L. JENSEN

electron toward the surface was far too optimistic, even though it worked well when the photon energy was comparable to the barrier height above the Fermi level. Rather, the transmission probability (expressed in terms of an energy argument) should be

 T ðEx þ
hoÞ ) T ðE þ
hoÞcos2 y ; ð675Þ where y is the angle of the electron trajectory with respect to normal. Another approximation used in the development of the thermal and field emission equations that must be modified in the case of photoemission is that the final state of the electron after absorption Figures into the analysis. Consequently, for the distribution of electrons in the current density, the replacement fFD ðEÞ ) fFD ðEÞf1  fFD ðE þ
hoÞg

ð676Þ

is made, in which the occupancy of the final state of the electron matters as to what electrons can be excited. Last (but not least), an electron of a given final‐state energy must transport to the surface without suffering a debilitating collision, and so the scattering factor as a function of energy must be present. That is, the factor fl ðcosy; E þ
ho Þ ¼ pð E Þ 

cosy cos y þ pðE þ
hoÞ md hkðE ÞtðE Þ


ð677Þ

is included in the integrand, where d is the laser penetration depth, pffiffiffiffiffiffiffiffiffiffi kðE Þ ¼ 2mE =
h, and the relaxation time t(E) has by now become all too familiar. Define 0 13=2 ð1 ð p=2 2m Mn ðks Þ ¼ ð2pÞ3 @ 2 A E 1=2 dE sinydyðks Þn
ð678Þ h
0 0 T fðE þ
hoÞcos2 ygfl ðcosy; E þ
hoÞfFD ðEÞf1  fFD ðE þ
hoÞg where particles traveling away from the surface (y > p/2) have been excluded and the following definitions for parallel ks ! kz and transverse ks ! kr momentum components are defined by
2 k2z h ¼ Ecos2 y 2m
2 k2r h ¼ Esin2 y 2m

ð679Þ

ELECTRON EMISSION PHYSICS

259

Thus, in Eq. (678), a sin2y term in the integrand raised to the power n/2 is recognized as the longitudinal momentum to the nth power. By way of contrast, if emittance were the focus, then the cos2y would be replaced by sin2y in the integrand to examine the transverse momentum moments. It is readily seen that Eq. (678) is far different than the modified Fowler– Dubridge approach, but it is also seen how the Fowler–Dubridge approximation is a consequence for photon energies not much in excess of the barrier height; the transmission probability in such cases only admits electrons fairly well pointed at the surface at the outset, and the integrand tapers off quickly for larger values of y. The solving of Eq. (678), however, is a rather protracted problem for which the energy, field, and temperature dependence, especially of the relaxation time embedded in p(E), the transmission probability T(E,) and the FD distribution function, make a numerical approach all but inevitable. Nevertheless, the leading‐order behavior is instructive to ascertain, and it is obtained by making the zero‐temperature, small‐field approximation. The former turns the FD distributions into step functions; the latter does the same with the transmission probability. The moments approach to the evaluation of current density (for reasons to be seen over time, the current‐density calculation is easier to consider than emittance) is obtained from the first moment of the distribution function for kz, for which

ARLD Jo ¼ 2 2 kB

ðm

ð1 EdE

mþfo

x2 dx; x þ pðE þ
hoÞ

ð680Þ

jðEÞ

where the energy ratio j(E) has been introduced and is defined by


jðE Þ 

mþf Eþ
ho

1=2

:

ð681Þ

Clearly, photoemission does not occur unless j(E) < 1; that is, the final electron energy exceeds the barrier maximum. Note that barrier factor f is used rather than the work function F as the low‐field approximation manifests itself as rendering the transmission probability to be a step function independent of whether the Schottky barrier–lowering factor is included or not. Using the approximation

260

KEVIN L. JENSEN

8 9 < = x2 d 1 þ ð1  pÞd  d2 dx ¼ p2 ln 1  : 1 þ p; 2 1d x þ p

ð1

  d þ O d2  1þp

ð682Þ

then Eq. (680) becomes Jo  2

ARLD k2B

n

ðm mþf
ho

1  jðE Þ3

o

1 þ pðE þ
hoÞ

EdE:

In turn, the leading‐order approximation to Eq. (683) is given by ( ) ARLD ð
ho  fÞ2 ð3m þ f 
hoÞ Jo  2 2 : 12ðm þ fÞ½1 þ pðm þ
hoÞ kB

ð683Þ

ð684Þ

Not unexpectedly, the dependence on the factor ð
ho  fÞ2 , anticipated from the modified Fowler–Dubridge approach, is prominent, but even more can be said. Compare Jo with a ‘‘current’’ that accounts for all the excited electrons and directed at the by the relation ð surface defined ð1 ARLD m ARLD EdE dx ¼ 2
hoð2m 
hoÞ: ð685Þ Jmax ¼ 2 2 kB m
ho kB 0 It immediately follows that for f <
ho < m Pð
ho; bF ; bT Þ 

Jo ð
ho  fÞ2 ð3m þ f 
hoÞ : ¼ hoð2m 
hoÞðm þ fÞ½1 þ pðm þ
hoÞ Jmax 6


ð686Þ

Eq. (686) seems rather far from the Fowler–Dubridge model. However, if the photon energy is approximately equal to the barrier height above the Fermi level, then Pð
ho  f; bF ; bT Þ 

ð
ho  fÞ2 ; 4m2 ½1 þ pðm þ fÞ

ð687Þ

an expression which, apart from a factor comparable to 2 to 4, very closely resembles the product of the scattering factor Fl and Fowler–Dubridge probability ratio fU ðbð
ho  fÞÞ=U ðbmÞg. Rather than succumb to such a temptation, however, the moments‐based Eq. (686) is used in preference to the modified Fowler–Dubridge‐like Eq. (687) in calculations of QE below in cases where an analytical approximation is used, and so the moments‐based approach identifies the total photo‐field‐thermal current as

261

ELECTRON EMISSION PHYSICS

Cu (SLAC)

Quantum efficiency [%]

10−1

10−2 Experimental Φ = 4.31 Φ = 5.10 60−40

10−3 200

220 240 Wavelength (nm)

260

280

FIGURE 86. Comparison of experimental data (circles) with theory (all parameters from literature sources). Assuming that the surface is composed of two crystal faces in 60/40 proportion, the solid blue line results (weighted average of the 4.31‐eV and 5.10‐eV lines). (Experimental data courtesy of D. Dowell, SLAC.)

Je ð
ho; F ; T Þ ¼

q ð1  RðyÞÞPð
ho; bF ; bT ÞIl þ JGTF ðF ; T Þ;
o h

ð688Þ

where JGTF(F,T) is the general thermal‐field contribution if tunneling and/or thermal emission are appreciably present. The photocurrent expression in Eq. (688) has required extensive calculation. Its historical development and performance has been cataloged in the literature (Jensen, 2003a; Jensen, Feldman, and O’Shea, 2003; Jensen, Feldman, Virgo, and O’Shea, 2003a,b; Jensen, Feldman, and O’Shea, 2004, 2005; Jensen, O’Shea, Feldman, and Moody, 2006; Jensen and Cahay, 2006; Jensen, Feldman, Moody, and O’Shea, 2006a,b; Jensen, Lau, and Jordan, 2006) as it was systematically tested in the treatment of bare metals, coated surfaces, and progressively more complex systems. Its evaluation requires a full‐fledged numerical solution to account for time dependence, temperature, scattering factors, reflectivity, and other explicit and/or implicit quantities that are otherwise carefully hidden in the folds of such a deceptively unassuming equation. In the following text, QE shall be the numerically evaluated ratio between total emitted charge and total incident energy as per Eq. (375), where the emitted charge is the time integral over a current density [Eq. (688)] for a uniformly illuminated area, and the total incident energy is the time integral over laser intensity over the same area wherein the laser profile is presumed to be Gaussian in time and uniform in space.

262

KEVIN L. JENSEN

(a)

(b)

FIGURE 87. (a) Surface of a sintered tungsten dispenser cathdode, showing evidence of crystal face variation, pore and profilimetry, and surface roughness as a consequence of machining. (b) Same as (a) but at a lower magnification. (Photographs courtesy of N. Moody (UMD/LANL).

L. The Quantum Efficiency of Bare Metals Metal photocathodes are common photocathodes: being relatively simple by comparison to photocathodes using low–work function coatings (necessary to significantly enhance the QE) in addition to being desirably rugged (although they require cleaning; Schmerge et al., 2006), metal photocathodes are natural testing grounds for the quality of the theoretical models that have so far been developed. The prerequisite factors to evaluate the photoemission

263

ELECTRON EMISSION PHYSICS

Work function [eV]

5.2

Cu

Nb

W

Mo

4.8

4.4

4 100 110 111 112 113 114 116 310 332 Crystal plane FIGURE 88. The work function of various crystal faces for several metals topical to photocathodes for the accelerator community.

QE(l)/QE (190 nm)

100 10−1 10−2 10−3 Copper

10−4 10−5 10−6

110 Face only 100 + 110 + 111 Face

210

240 270 Wavelength [nm]

300

FIGURE 89. Difference in quantum efficiency for a pure (110) face compared to a surface equally composed of the 100, 110, and 111 faces of copper.

current from simple metals have been described, from the reflectivity and laser penetration depth to the dependence of the scattering factors on temperature and finally to the probability of emission. Several sources are available for comparisons. As a first comparison, consider the measured QE of copper (keeping in mind the caveats about DOS), as a function of wavelength before and after cleaning with a hydrogen ion beam, shown by Dowell et al. (2006) in their

264

KEVIN L. JENSEN

Figure 2 for the line designated ‘‘10.23 mC’’; in this line, it was argued that the contamination that had collected on the surface was removed—albeit not entirely, as a residual 8% of the surface was claimed to be covered with carbon (a high–work function material as a contaminant). Analogous results were obtained by Moody et al. (Moody, 2006; Moody et al., 2007) for the cleaning of tungsten with an argon ion beam. A comparison of the Dowell et al. data with a simulation based on Eq. (688) with all quantities such as relaxation time, reflectivity, and other embedded factors calculated using the models of Sections II and III is shown in Figure 86, where the intensity of the incident light is so low as to not make demands on the numerical calculation of a temperature rise. Various sources of differences and errors make a comparison to experimental data somewhat of an art. In assessing the performance of the theory, it is important to estimate these effects and the comparative magnitude of change that they would entail. They are variations of work function with crystal face, differences in the DOS compared to the nearly free electron gas approximation, effects of surface structure and/or reflections and/or field enhancements, and the impact of contamination. The existence of so many seemingly suggests that agreement between the theory herein and actual data bears a serendipitous relationship, but that would be an overly cynical insinuation. The various complications create changes that can be at odds (in the direction for which they modify the model above) with each other and, moreover, do not result in large multiplicative factors. In actuality, the success of the moments‐based emission model is notable. Moreover, there are factors that, if not unknown, are unknowable and therefore must be accounted for by other means (e.g., surface profilimetry; Jensen, Feldman, Virgo, and O’Shea, 2003b; Jensen, Lau, and Jordan, 2006), which will tend to result in an ‘‘effective’’ field enhancement factor to account for surface roughness, and geometrical features complicate transport near the surface in complex ways (Mayer and Vigneron, 1997). Most important, surface conditions are not static; they are affected by the migration of coatings across metal surfaces, evaporation rates, degradation effects, and performance characteristics (Jensen et al., 2007). Complications to the simple metals are considered here, deferring until the next section the significantly more complicated impact of surface coatings on all manner of electron emission effects. 1. Variation of Work Function With Crystal Face Particularly in studies of thermionic emission from dispenser cathodes, where sintered metals such as tungsten expose a number of crystal faces (an example is shown in Figure 87) on which low–work function coatings rest, it has long been appreciated that the consequences of the presence of different exposed faces translated into different emission current densities (Adler and Longo, 1986; Haas and Thomas, 1968;). Figure 88 shows the variation of work

265

ELECTRON EMISSION PHYSICS

3 U bT hw −Φ + 4QsF

Ratio

2.5

2

U bT hw −Φ + 4QF

s=3 T = 300 K F = 1 MV/m

1.5

1

200

220 240 Wavelength [nm]

260

280

FIGURE 90. Behavior of the Fowler–Dubridge function as a consequence of field enhancement for a hemisphere.

function for typical faces for a few thermionic, field emission, and photoemission metals. The cause of the different work functions for various faces, according to Brodie (1995), relates the crystal face work function to dimensions of the underlying atoms and the effective mass with respect to Fermi energy of the electron along different crystal planes in bulk. Consider the expected photoemission differences if the first three crystal faces of copper (100, 110, and 111 with F ¼ 5.1, 4.48, and 4.94, respectively) are present in equal proportion on a surface, as compared to a monocrystalline surface of the 110 face, the results of which are shown in Figure 89 under the assumption that impact of reflectivity and scattering are more or less equal, and by using the Fowler–Dubridge representation for the escape probability, as it is easier to implement and is approximately correct. By taking the ratio with a reference QE at l ¼ 190 nm, factors common to all three faces cease to figure into the estimate. There are reductions in the QE that vary as a function of wavelength as the photon energy drops below the various crystal face work functions as the wavelength increases; consequently, the agreement is better for shorter wavelengths (the QE for 3 faces is approximately 70% of the QE for the 110 face at 190 nm) than it is for longer wavelengths (the QE for 3 faces is 1/3 of the QE for the 110 face for wavelengths longer than 270 nm, where the factor of 3 represents the assumption that the crystal faces are equally represented).

266

KEVIN L. JENSEN

100 Copper 90 Reflectivity [%]

Lead 80 70 60 50 40

0

20

40 60 Angle [deg]

80

FIGURE 91. Reflectivity of copper and lead as a function of incidence angle.

FIGURE 92. The surface of solid lead. The white square is 4 mm on a side. (Photograph courtesy of J. Smedley, Brookhaven National Laboratory.)

267

ELECTRON EMISSION PHYSICS

FIGURE 93. The surface of magnetron‐sputtered lead. The white square is 5 mm on a side. (Photograph courtesy of J. Smedley, Brookhaven National Laboratory.)

r1

r2

1-r1-r2

FIGURE 94. Surface roughness and its relation to increased absorption, as multiple hits on the surface increase the absorption probability.

268

KEVIN L. JENSEN

2. The Density of States With Respect to the Nearly Free Electron Gas Model The nearly free electron gas model has formed the basis of most models described herein due to its simplicity and easy explanatory value. In contrast, proper calculations should use the correct DOS in three dimensions for the actual metals under question. There are substantial differences between the simple metals and the transition metals for which the narrow d band fills and for which the noble metals (a surreal title given the passions they arouse), such as gold, silver, and copper, have completely filled d bands (Sutton, 1993). A proper account of electron emission (Modinos, 1984) and in particular photoemission (Berglund and Spicer, 1964a,b; Dowell et al., 1997; Ishida, 1990; Janak, 1969; Krolikowski and Spicer, 1969) pays attention to the empirical DOS or uses sophisticated theoretical methods to estimate the DOS, especially for the transition metals. Alternately, dedicated sites for the calculation of the DOS for various elements exist (‘‘NRL Electronic Structures Database.’’ http://cst-www.nrl.navy.mil/ElectronicStructureDatabase). Calculating the DOS, however, requires an understanding of the underlying crystal structure, is nontrivial, and requires that fairly substantial theoretical methods be brought to bear; the case for copper is a particularly complicated one (Campillo et al., 2000)—the repeated use of copper as a case study herein is therefore not without a bit of irony. It induces complexity far beyond the nearly free electron gas model that provides a useful simple model for

z q

ρ

Perspective

dr

Top-down

FIGURE 95. Relation of the incidence angle to the differential surface area for the model of a hemisphere.

ELECTRON EMISSION PHYSICS

269

example cases we have considered. This most important of modifications to the photoemission models is therefore relegated to the in‐depth treatments of the literature. 3. Surface Structure, Multiple Reflections, and Field Enhancement In the moments‐based model behind Eq. (678), changes induced by surface structure are not explicitly accommodated. These manifest themselves in two ways: as a field enhancement changing of the emission barrier, and as a change in the reflection of the incident light due to a crystal face being at an angle to the incident light. Consider field enhancement first, and as a pedagogical example, consider a hemispherical bump (a ‘‘boss’’). Field enhancement tends to change over a bump; recall that Schottky barrier–lowering for a field of 1 MV/m is on the order of 0.04 eV, which, while not great, can affect estimates of QE. Letting s represent the field enhancement factor compared to a flat surface, then the approximate increase in QE may be estimated from the modified Fowler–Dubridge formula as pffiffiffiffiffiffiffiffiffiffiffiffi Jo ðsF Þ U½bT ð
ho  F þ 4QsF Þ pffiffiffiffiffiffiffiffiffiffi :  ð689Þ J o ðF Þ ho  F þ 4QF Þ U½bT ð
The behavior of Eq. (689) for copperlike parameters is shown in Figure 90. The impact of field enhancement is offset by the areas involved over which the enhancement factor is significant. Using the example of the hemisphere, while the enhancement on‐axis is a factor of 3, the effective area over which this occurs is dA ¼ 2pr2 sinydy, and therefore, smaller areas contribute near the axis where the enhancement is strong (similar arguments are at work in the definition of the emission area of a field emitter (Forbes and Jensen, 2001, for example). Next consider the change in reflectivity, again for the boss example as the angle the incident light makes with the normal to the surface is equivalent to the polar angle measured from the apex. The reflectivity is dependent on the particular material; the cases for copper and lead are shown in Figure 91. The reflectivity does not change appreciably until past 60 degrees, at which point it begins to climb to unity. Such an effect offsets the greater emission area associated with the rings of larger diameter over which the reflectivity is constant. Thus, the centermost parts of the hemisphere contribute the most to the QE of a rough surface, although the effective emission area is less than suggested by the dimensions of the illumination area. To gain an appreciation of the complexity of the physical surfaces, consider the cases of solid and magnetron sputtered lead surfaces, images of which (taken by J. Smedley, BNL) are shown in Figures 92 and 93, respectively.

270

KEVIN L. JENSEN

Both appear smooth on a macroscopic scale, but micron‐scale resolution shows just how complex the surfaces are, particularly the magnetron sputtered example, which evinces greater QE than the solid lead surface (in the latter, the sharp eye will notice sandlike grains pressed into the lead surface, which are residual grains from diamond polishing). The canyonlike complexity of these surfaces suggests yet another possible effect—the probability that light reflected from the side of a protrusion, rather than being sent on its way from the surface, is rather sent to strike another region on the surface, as suggested in Figure 94. With high reflectivity a disproportionately greater impact results. QE depends on the amount of light absorbed, so the question arises as to how much more light is absorbed when multiple reflections are present. Let the proportion of the surface accounting for one reflection be r1, that for two reflections be r2, and assume that all photons experiencing more than two reflections are in fact absorbed. The increase in QE will then be, to a first approximation, the ratio of photons absorbed on a rough surface on regions where one, two, and more than two reflections occur compared to the condition where only one reflection occurs. The number of photons absorbed from those incident on region 3 is unity by assumption; the number absorbed incident on region 1 is (1 – R); and the number absorbed incident on region 2 is R(1 – R). Therefore, the ratio of the number of absorbed photons for the rough surface compared to the smooth should approximately behave as QErough ð1  r1  r2 Þ þ ð1  RÞr1 þ Rð1  RÞr2 1  Rr1  R2 r2 ¼ :  QEsmooth 1R 1R

ð690Þ

For a rather stylized example, if the three regions are equally represented and the reflectivity is 75%, then the improvement is 9/4 ¼ 2.25. A complication is the fact that the reflectivity generally depends on incidence angle and Eq. (690) presumes the reflectivity to be more or less constant. The question arises, then, as to how much the variation in reflectivity will affect matters; its impact will be to reduce the effective absorbing area as surfaces faceted away from the normal to the macroscopic surface will subject incoming photons to more oblique incidence angles, as suggested in Figure 95. For uniform intensity light incident from the top, the boss will be ‘‘seen’’ as a circle (the ‘‘top‐down’’ perspective) so that the intensity of light Io illuminating each ribbon defined by 2prdr will be the same, even though the intensity Io cosy on the actual surface ribbon 2pa2 sinydy, where a is the radius of the boss and r ¼ asiny, diminishes as y increases. The product of the reduced intensity and the increased ribbon area pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pr dr2 þ dz2 ¼ 2prdr=cosy offset each other, resulting in an integrand that puts the work of the y‐variation only on the reflectivity factor. Therefore, in the modified Fowler–Dubridge model, the ratio of the QE for an

271

ELECTRON EMISSION PHYSICS

(a)

October 31, 2001

1 0.95

600

0.90

y pixel

550

0.85 0.80

500

0.75 450

0.70 0.65

400

0.60 350

0.55 350

400

y pixel

(b)

450

500 x pixel

550

600

November 04, 2001

1.0

600

0.9

550

0.8 0.7

500

0.6 450 0.5 400

0.4

350

0.3 350

400

450

500 x pixel

550

FIGURE 96. (Continues)

600

272

KEVIN L. JENSEN

(c)

December 04, 2001 1.0 0.9

600

0.8 550

y pixel

0.7 0.6

500

0.5 450 0.4 0.3

400

0.2 350

0.1 350

400

(d)

450

500 x pixel

550

600

December 10, 2001

y pixel

1.0 600

0.9

550

0.8 0.7

500

0.6 450 0.5 400 0.4 350

0.3 350

400

450

500 x pixel

550

600

650

FIGURE 96. (a) Quantum efficiency plots of the APS Mg LEUTL Photocathode: pixels are approximately 10 mm on a side. This image is before any cleaning. (b) Same as (a) but after first cleaning. (c) Same as (b) but 1 month later, showing degradation from operation. (d) Same as (c) but after second cleaning. Uniformity has been improved and contamination reduced, but the pattern has evolved from (b). (Data for all images courtesy of J. Lewellen, Argonne National Laboratory.)

273

ELECTRON EMISSION PHYSICS

illuminated boss compared to a uniformly illuminated disk of the same radius is given by ða f1  R½yðrÞg2prdr QEboss  0ð a QEdisk ð1  Rð0ÞÞ2prdr ð691Þ ð p=20 f1  RðyÞgsinð2yÞdy 0 ¼ ð1  Rð0ÞÞ For copper, where R behaves (to a good approximation) as h p i RCu ðyÞ  Ro þ ð1  Ro Þexp mo  y ; 2

ð692Þ

with Ro ¼ 0.49775 and mo ¼ 6.2329, the evaluation of Eq. (691) is analytic and results in QEboss  0:95338QEdisk , a difference that is not eye‐catching for a hemisphere but which, when applied to the multifaceted structures perhaps analogous to the magnetron sputtered surfaces, might have consequences of greater significance. 4. Contamination and Effective Emission Area The problem of cleaning contamination from metal surfaces to expose clean

100

QEpatch/QEbare [%]

7% 90 15% 80

30%

70 200

220

240 260 Wavelength [nm]

280

300

FIGURE 97. Changes in effective quantum efficiency for a surface partially covered with a higher–work function material (percentages indicate degree of coverage of said material).

274

KEVIN L. JENSEN

crystal faces is well known (Haas and Thomas, 1968). In the case of thermal‐ field emission from refractory metals such as tungsten, a grueling heating of the needles can be performed to drive off all manner of contaminations (and also allow for deformation of the emitter tip as a consequence of the balancing of surface tension and field; Barbour et al., 1960). Such techniques cannot be used with microfabricated field emitters (field emitter arrays) or the metals generally used as photocathodes because the temperatures required are far too high for the materials used or close to the melting point of the favored metals (such as copper, which has a melting point of 1358 K), but contamination is still problematic. Although Spindt et al. discussed molybdenum field emitters, their description of the condition of emitter tips (from chapter 4 of Zhu, 2001) is an elegant summary for metal surfaces in general: ‘‘. . .microfabricated emitter arrays are rarely heated for cleaning at more than 450 C, and this is not sufficient to produce an atomically clean surface. As a result, we find ourselves working with an ill‐defined emitting surface that can probably be best described as a combination of several microcrystalline surfaces, grain boundaries, and adsorbates. In addition, it is a dynamic situation as adsorbates diffuse about the surface and the surface evolves toward equilibrium with its environment.’’ [Spindt et al., in chapter 4 of Zhu, 2001)]

For photoemitters, methods other than, or in addition to, heating must be used to reduce the impact of adsorbates and contamination/degradation. One such method is to subject the metal surface to a laser beam focused to an intensity just below the damage threshold of the metal and then scan the surface, a process that alters the surface as revealed by changes in the emission pattern (Girardeau‐Montaut, Tomas, and Girardeau‐Montaut, 1997; Smedley, 2001 Srinivasan‐Rao et al., 1998; Tomas, Vinet, and Girardeau‐ Montaut, 1999). More recently, methods of cleaning the surface using hydrogen (Dowell et al., 2006) and argon (Moody, 2006; Moody et al., 2007) have proven quite successful at cleaning and restoring a metal surface to initial QE values. Even so, in the hydrogen ion cleaning of copper, as Dowell et al. point out, the data suggest that 7% of the surface retains some carbon coverage; since the work function of carbon is high, this suggests that a fraction of a cleaned surface nevertheless does not contribute and therefore gives the appearance of a lower overall QE than would otherwise be the case, a problem conceptually very similar to the impact of nonuniformity (see Chapter 2 of Herring and Nichols, 1949) or of poisoning of a low–work function coating on the surface of a thermionic emitter of particular concern to the dispenser cathode community (Marrian and Shih, 1989). An example of the changing of the QE associated with an actual metal photocathode is shown in a series of QE measurements of the magnesium photocathode used for the advanced photon source (APS) low‐energy undulator test line (LEUTL) at Argonne National Laboratory (Lewellen and

ELECTRON EMISSION PHYSICS

275

Borland, 2001; Lewellen et al., 2002) taken by J. Lewellen (from ANL) over a few months. In the sequence of images shown in Figure 96, two trends are apparent. First, the ‘‘cleaning’’ of the photocathode by a laser significantly improves the QE, both in magnitude and uniformity (the images were scaled to their maximum value in each case and therefore show relative, not absolute, performance so as to accentuate contrast—therefore the apparent QE of one image does not correspond to the QE of another, although the scale band to the right does indicate the relative magnitude within the image). Second, recleaning the surface does not return the cathode to its initial state, although it does improve matters; changes to the surface accompany the cleaning process. One such change, not apparent, is the increase in dark current after the cleaning process, an indication of changes in surface structure and geometry—that is, cleaning ‘‘roughens up’’ the cathode. This affects emittance, field emission, and the like—apart from the impact of changes in work function due to removal and redeposition of contamination and adsorbates. If a region of a photoemitting surface experiences conditions such that it exhibits a higher work function than surrounding areas (due to contamination, crystal face, or another effect), then the overall QE is reduced. If an area dA of a total area A exhibits a work function of F þ dF, then, compared to a bare (or uniform) surface, the QE becomes, as estimated by the modified Fowler–Dubridge model,
 QEpatch dA U½bT ð
ho  f  dFÞ ðA  dAÞ : ð693Þ  þ A U½bT ð
A QEbare ho  fÞ For photon energies well above the barrier, the ratio is close to unity, but for energies closer to the barrier, the reduction in QE is approximately proportional to the uncontaminated proportion of the surface. As an example, consider the contamination to have a work function comparable to polycrystalline carbon of approximately 5.0 eV. The impact on QE as a function of wavelength is shown in Figure 97 for coverages of 7%, 15%, and 30%. M. The Emittance and Brightness of Photocathodes The moments‐based formalism used to determine the QE of bare metals can now be used to determine the emittance associated with photoemission. The need for such descriptions arises, for example, in the effort to provide physics‐based emission models needed by advanced simulation codes (Lewellen, 2001; Petillo et al., 2005; Travier et al., 1997; Zhou et al., 2002). The tacit assumption underlying the present description is that the electron beam is used to convert spontaneous electromagnetic radiation to coherent

276

KEVIN L. JENSEN

radiation from a beam‐wave interaction characteristic of a broad class of vacuum electronic devices (Abrams et al., 2001; Gilmour, 1986; Parker et al., 2002). There are other uses of electron sources, but it is the VE applications in particular that have provided the fundamental paradigm for demands on the electron source that drives the present discussion. Traditionally, thermionic cathodes are used in or sought for microwave and power amplifiers (cold cathodes have also been considered and used (Makishima et al., 1999; Whaley et al., 2000), whereas photocathodes are the source of choice for the ‘‘big dogs’’ of advanced RF photoinjectors for high‐power free‐electron lasers—and interesting combinations of photo and field emission may enable ‘‘little dogs’’ based on photostimulated needle cathodes (Brau, 1997, 1998; Jensen, Feldman, Moody, and O’Shea, 2006b; Lewellen and Brau, 2003). The two concepts that determine the quality of an electron beam introduced in Section II.I (Thermal Emittance) are emittance and brightness. They are of such paramount importance for the accelerator and vacuum electronics communities that even an extended description in the confines of the present treatment would only scantily cover the literature (the canonical texts of Reiser, 1994, and Humphries, 1986 and 1990, are general treatments, but see also Abrams et al., 2001; Anderson et al., 2002; Brau, 1997; 1998; Carlsten et al., Fraser and Sheffield, 1987; 1988; Fraser et al., 1985; Humphries, 1990; Jensen, O’Shea, Feldman, and Moody, 2006; O’Shea, 1995, 1998; O’Shea et al., 1993; Parker et al., 2002; Rao et al., 2006; Reiser, 1994; Rosenzweig et al., 1994; Travier, 1991; Tsang, Srinivasanrao, and Fischer, 1992). Thermionic emittance was treated before (Eq. (373)); extending that treatment to photoemission is the present objective. FELs represent ‘‘tunable’’ sources of narrow‐band light—wavelengths in the hundreds (UV) to the tens (XUV) of nanometers—and megawatt‐class devices may be possible if the brightness of the electron source can be improved, and RF photocathodes appear to be the most likely source capable of doing so (O’Shea et al., 1993, 1995). The average power of the FEL is limited by the electron beam average power and beam brightness, for which the improvements entailed by photocathodes literally outshine the thermionic cathode competition (Dowell et al., 1993) in terms of beam brightness. In an FEL, a pulse train of electron bunches is created, each containing a substantial amount of charge (on the order of 0.1 to 2.0 nC). Overlap of the lightwave field with the electron bunch is critical for gain, and that entails a tolerable upper limit on the transverse emittance that can be endured (Fraser and Sheffield, 1987) (longitudinal emittance, another concern, is not discussed here). Beams with higher current and smaller emittance enable shorter wavelength and more powerful FELs. Electrons outside the laser beam do not contribute much to the coherent radiation and are wasted, or—what may be worse—electrons outside the core beam (generally called halo) cause some damage elsewhere where it is ill

277

ELECTRON EMISSION PHYSICS

tolerated (Bohn and Sideris, 2003). The extension of the derivation of the emittance of a thermionic source to a photocathode requires more effort (Jensen, O’Shea, Feldman, and Moody, 2006) than the pleasingly (and relatively) uncomplicated derivation for thermal emittance.

(a) 4

e n,rms [mm-mrad]

Cs on Cu Φ = 1.8 eV

Numerical Eq. 699

3

Cu Φ = 4.5 eV

2

1 Thermal 0 200

300

400 500 Wavelength [nm]

600

700

(b) 4 3 109

Bn [A/cm2]

3 3 109

2 3 109

1 3 109

Cu Cs on Cu

0 200

300

400 500 Wavelength [nm]

600

700

FIGURE 98. (a) Comparison of the analytical model of emittance [Eq. (699)] with its numerical evaluation. (b) Estimates of brightness based on Eq. (700) for copper and cesium‐coated copper (work functions of 4.5 and 1.8 eV, respectively).

278

KEVIN L. JENSEN

The low‐temperature limit of Eq. (678) and the Richardson (step function) approximation to the transmission probability result in the approximation 3ðnþ1Þ=2 3ðnþ3Þ=2 2 ð1 2m m 4 4 ð
þ x  15 ho  f Þ 5 Mn ¼ h  fÞ h2 ð2pÞ2
0 ð
2 3 0 1 2

1

1 A 6
G4p½ðm þ fÞð1 þ DxÞ; @ 1 þ Dx

1=2

ð694Þ

n7 ; 5dx 2

where the dimensionless quantity D has been introduced and defined by D¼


o  f h : mþf

ð695Þ

For photoemission conditions such that the photon energy is not much larger than the barrier height, then D can be small for metals. The new function G is defined by Gða; b; sÞ 

ð1

s

sþ1

xð1  x2 Þ ð 1  b2 Þ dx  ; 2ð s þ 1Þ ð 1 þ aÞ b ð x þ aÞ

ð696Þ

where the RHS is an approximation rather than an exact result. In fact, when

(a) 3.0 Cu l = 266 nm Φ = 4.5 eV

e n,rms [mm-mrad]

2.5 2.0 1.5

Cs on Cu l = 400 nm Φ = 1.8 eV

1.0 0.5

1

10 Field [MV/m] Numerical

100

Eq. (699)

FIGURE 99. (Continues)

279

ELECTRON EMISSION PHYSICS

(b) 6 ⫻ 109

Bn [A/cm2]

5 ⫻ 109 Cs on Cu l = 400 nm Φ = 1.8 eV

4 ⫻ 109 3 ⫻ 109

Cu l = 266 nm Φ = 4.5 eV

2 ⫻ 109 1 ⫻ 109

1

10 Field [MV/m]

100

(c) Cs on Cu: l = 400 nm, Φ = 1.8 eV

e n,rms [mm-mrad]

3

Numerical Eq. (699)

2

Eq. (373)

1 Cu: l = 266 nm, Φ = 4.5 eV 400

600

800 1000 Temperature [K]

1200

1400

FIGURE 99. (a) Comparison of the numerical evaluation of emittance using the moments to the analytical formula of Eq. (699) for copper and cesium on copper. (b) Brightness as evaluated using Eq. (700) The brightness for bare copper has been multipled by a factor of 10 so as to allow for a visual comparison. (c) Numerically evaluated photoemittance compared to analytical model: the latter is temperature‐independent, causing the numerical evaluation (which is dependent on T) to diverge at larger temperatures. Also shown is the thermal emittance, Eq. (699).

s is an integer, an exact result can be found, namely,

280

KEVIN L. JENSEN

IV. LOW–WORK FUNCTION COATINGS AND ENHANCED EMISSION A. Historical Perspective In the presumptively halcyon days of the 1920s and 1930s, when the equations of electron emission physics were born from the marriage of quantum mechanics and statistical mechanics, much effort was devoted to understanding emission, characterizing work function, ferreting out the impact of different crystal faces, and assessing the consequences of the absorption of materials such as cesium and thorium on metals with the tendency to increase emission current. Many of the great names of physics left their mark in disparate fields from which the literature on electron beams trace their origins. At about the same time, following the pioneering research of both Heinrich Hertz and Nikola Tesla, Albert Taylor and Leo Young at the U.S. Naval Research Laboratory (NRL)3 demonstrated (both by accident and intent) the first continuous‐wave (CW) radar system that another NRL scientist, Robert W. Page, succeeded in transforming into a pulsed radar system in the early 1930s (Allison, 1981; Kevles, 1987). The onset of war accelerated matters considerably. In response to the urgent need of the United Kingdom for radar systems, a magnetron developed by the U.K. scientists John Randall and Harry Boot demonstrated enough power to make radar practical (Osepchuk and Ruden, 2005; decades later, magnetrons filled another, albeit more benign need for microwave ovens). Klystrons were developed by the Varian brothers in 1937 (Tallerico, 2005), and the traveling wave tube was invented by the U.K. scientist Rudolph Kompfner and later refined by him and John R. Pierce at Bell Labs in the United States (Lerner and Trigg, 1991). Were it not for the political events of that time that held history in thrall, the intellectual ferment was likely exhilarating. By the 1940s, the technologies made possible by harnessing electron emission for vacuum tubes came to be recognized by those who understood how the capabilities could be used to advantage in the pressing global conflicts of the time.4 Nothing spawns innovation and investment quite like a convergence of military and commercial interests. The needs of radar, communications, electronic warfare (Granatstein and Armstrong, 1999), and directed‐energy devices 3 Coincidentally, thermionic emission (referring to the emission of ‘‘thermions’’) was originally designated the ‘‘Edison effect’’ after Thomas Edison, who went on to champion the creation of the U.S. Naval Research Laboratory and who is that laboratory’s patron saint. The history of NRL and radar development there is detailed in Allison (1981). 4 Power tubes refer to early magnetrons, klystrons, traveling wave tubes, and later gyrotrons and free electron lasers; this is a peculiar appellation as there is nothing glass tube–like about them.

ELECTRON EMISSION PHYSICS

281

(Bennett and Dowell, 1999; O’Shea and Bennett, 1997) for the capabilities of vacuum devices led to rapid advances on several fronts. Radar and vacuum electronic research at NRL itself and many other institutions in the United States and worldwide became vigorous for several decades and remains an area of active research. While solid‐state technology applied to radar has made astounding advancements in a moderately shorter time, for high‐power applications the playing field still belongs to the ‘‘tubes’’ (Abrams et al., 2001; Freund and Neil, 1999; Granatstein, Parker, and Armstrong, 1999; O’Shea and Freund, 2001). Of the five technologies necessary for the maturation of RF vacuum technology (Parker et al., 2002)—namely, the linear beam, periodic permanent magnet focusing, the depressed collector, the dispenser cathode, and the metal/ceramic packaging—the innovation that is of present concern is the fourth: the dispenser cathode. Its ubiquitous presence in all manner of devices, such as cathode ray tubes in displays, advanced radar systems, particle accelerators, satellite communications, electronic warfare systems, microwave generators, attests to its sweeping importance. In time, other cathodes offering other capabilities came to the fore, but the idea of lowering the work function of a material through the selective application of materials has captured the attention not only of the dispenser cathode community, but also the field emission and photoemission communities as well as attested by more than 80 years of research. What happens on the surface of a metal when elements like cesium and barium come to roost is a protracted problem in surface science for which extensive treatments are to be found (Haas and Thomas, 1968; Modinos, 1984; Mo¨nch, 1995; Prutton, 1994; Sommer, 1968). Here we provide an account of the physics and its application to the interpretation of photoemission data from partially covered surfaces in a manner that uses what has come before and a theory of work function reduction developed by Gyftopoulos and Levine (Gyftopoulos and Levine, 1962; Jensen, Feldman, Moody, and O’Shea, 2006a,b; Jensen, Feldman, Virgo, and O’Shea, 2003b; Levine and Gyftopoulos, 1964a,b; Moody et al., 2007).

B. A Simple Model of a Low–Work Function Coating When an atom of cesium sits on the surface of a metal such as tungsten, its weakly bound outer electron easily transfers to the bulk material. The ion— or, perhaps more correctly, the polarized atom—left behind induces an image charge. A very trivial model of work function reduction is to then envision that a sheet of charged ions opposite of their image charges exists, looking very much like a capacitor. The surface charge density s and the distance of

282

KEVIN L. JENSEN

the partial ion to its image charge d allow for an estimation of the potential drop that can be interpreted as a reduction in the work function of the surface. Assume for sake of argument that the charge density is a fraction s of a unit charge for one atom over an area equal to the atomic diameter squared, or s ¼ sq=ð2rC Þ2 , where rC is the radius of the atom (for cesium, rC ¼ 0.52 nm), which suggests that d is larger than, but near, 2rC. It then follows from elementary considerations that
 2pafs
hc s DF
q s ð701Þ d¼ eo rC For cesium‐like parameters, the work function reduction is from 4.5 eV to 1.6 eV, or 2.9 eV, so that s is approximately 1/6, reinforcing the notion that the cesium atoms are more like polarized atoms, or dipoles, than ions, as suggested by simulations (Hemstreet and Chubb, 1993; Hemstreet, Chubb, and Pickett, 1989). If the surface coverage is not a monolayer (y ¼ 1) but rather exhibits fractional coverage (y < 1), then s
qy=rC and the work function decreases with reduced coverage. This would suggest that the overall work function decreases from bulk values to the monolayer coverage value as y increases from 0 to 1. What is observed is that for very low coverage values, the reduction is in fact roughly linear, but as monolayer coverage is approached, changes in y do not change the work function appreciably from its monolayer values. In other words, rather than DF being linear in y, it resembles a more complex function. The determination of that function is the goal of Gyftopoulos–Levine theory. C. A Less Simple Model of the Low–Work Function Coating From the late 1970s and thereafter, considerable industrial effort was devoted to understanding the operation of the dispenser cathode. Much was to be gained from a longer‐life, lower–work function cathode for military, space, and commercial applications, and a commensurate effort was devoted by industry and government to characterizing them, finding new candidates, and understanding the operation of these complex constructs. A small and pragmatic literature base aimed at studying the operational characteristics of these cathodes was published in the journal literature but also in the Technical Digest of the International Electron Devices Meeting (IEDM) and the Tri‐Service/NASA Cathode Workshop (see Adler and Longo, 1986; Chubun and Sudakova, 1997; Cortenraad et al., 1999; Falce and Longo, 2004; Gartner et al., 1999; Green, 1980; Haas and Thomas, 1968; Haas, Shih, and Marrian, 1983; Haas, Thomas, Marrian, and Shih, 1989;

ELECTRON EMISSION PHYSICS

283

Haas, Thomas, Shih, and Marrian, 1989; Jensen et al., 2003b; Jensen, Lau, and Levush, 2000; Jones and Grant, 1983; Longo, 1978, 1980, 2000, 2003; Longo, Adler, and Falce, 1984; Longo, Tighe, and Harrison, 2002; Marrian, Haas, and Shih, 1983; Marrian and Shih, 1989; Marrian, Shih, and Haas, 1983; Schmidt and Gomer, 1965; Shih, Yater, and Hor, 2005; Thomas, 1985; Vancil and Wintucky, 2006, for a representative cross section). A model by Longo, Adler, and Falce (1984) provides a concise account of the work function variation. Dispenser cathodes are developed by pressing small grains of tungsten together under heating (sintering). The joined grains are porous; the spaces between are filled with material that, when heated, liberates barium, which then migrates to the surface. In the operation of a cathode, barium diffuses to the surface and exudes from pores that are randomly spaced but generally such that the pore‐to‐pore separation is on the order of the grain size: 6–10 mm. Early in the life of the cathode, the barium arrival rate at the surface can exceed what is required for monolayer coverage. At most a monolayer of barium atoms builds on the surface as bulk evaporation rates are orders of magnitude faster than the monolayer evaporation rates, a consequence of the different bond strength between barium and itself compared to barium and tungsten (Forman, 1984). Nevertheless, given the pores, speculation that small islands of barium formed around them was a hypothesis worth investigating and so Longo, Adler, and Falce (1984) set out to assess what the average work function might be and the possible consequence(s) of island formation on the operational lifetime of dispenser cathodes. While more phenomenological than the Gyftopoulos–Levine theory, it captures some features rather easily. It is assumed that the work function of a surface is a weighted average between the work function of the bulk material (in this example, tungsten—W) and the work function of the coating (designated by a C; the coatings can change from barium to barium oxide to cesium to whatever, and so a generic designation is used). Measurements of the work function of partially coated surfaces exhibit a minimum, sometimes at values under a monolayer, as shown in Figure 100 for data adapted from figure 22 of Schmidt and Gomer (1965) for the metals cesium, potassium, barium, and strontium. Assuming for the moment that the crystal plane on which the coating rests is uniform (it need not be; unless a single crystal is used, coverage and work function will be affected by crystal face; the photographs in Schmidt and Gomer provide a rich catalog of images of differing coverage on different planes of a needle), then a fictitious ‘‘picture’’ of such a surface near a pore might well resemble Figure 101, which suggests regions about which there may be no coating (‘‘bare’’), a monolayer coating (‘‘monolayer’’), or many layers (‘‘multiple layers’’) for which the work function of that region looks like the bulk work

284

KEVIN L. JENSEN

Cs

Work function [eV]

4

K Ba St

3

2

0

2

4 n [1014 atoms/cm2]

6

FIGURE 100. Variation of work function with surface coverage for various coverings (surface density), based on figure 22 of Schmidt and Gomer (1965).

Bare Monolayer

Multiple layer

FIGURE 101. Representation of how the coverage near a pore on a dispenser cathode surface may appear in the Longo model.

function of C. This suggests that the ‘‘macroscopic’’ work function, as might be obtained from a Richardson plot, is a sum of differing terms of the form hFðyÞi ¼ fw Aw ðyÞ þ fc Ac ðyÞ;

ð702Þ

where the urge to interpret the A factors as areas is strong—but should be resisted as they are instead weights of a distribution. They should, however, have some relation to the actual areas of coverage, and Longo et al. (1984) suggests the appellation weighted areas. If they act like areas, then small

ELECTRON EMISSION PHYSICS

285

changes in the coverage will cause small changes in the average work function and will depend on the amount of each area so covered. Thus, one might expect @y Ac ðyÞ ¼ aAc ðyÞ @y Aw ðyÞ ¼ bfAw ð0Þ  Aw ðyÞg

ð703Þ

where a and b are rate constants,and y is the fractional monolayer coverage: y > 1 means more than a monolayer present, and Aw ð0Þ is a bare surface. Solutions to Eq. (703), when inserted into Eq. (702) and normalized to unit area, suggest that the average work function is then hFðyÞi ¼ eay fw þ ð1  eby Þfc ;

ð704Þ

an equation that properly expresses an intuitive feeling: when the coverage is low, then changes in the average work function appear to be linear in y but, depending on the values of the rate constants a and b, then Eq. (704) can exhibit a minimum at submonolayer coverage. Define ym to be the value of y that minimizes hFðyÞi, that is,

 a fw lim @y hFðyÞi ¼ 0 ) ð705Þ ¼ expfðb  aÞym g; y!ym b fc where Fðym Þ ¼ Fmin can be less than the bulk work function of the covering material. For example, barium on tungsten has a minimum work function of 2.0 eV, whereas the work function of bulk barium is 2.55 eV. For low coverage, the work function variation with coverage is almost linear, and so lim @y hFðyÞi  Sf ¼ afw  bfc : y!0

ð706Þ

From experimental data the variation of work function (for example, as shown in Figure 102 for barium) on the assumption that 4:3  1014 #=cm2 atoms constitutes a monolayer (as suggested by the scaling of Schmidt, 1967) and letting a=b
2, then the Longo approximation of Eq. (704) compares with the data of Schmidt in Figure 102; the Longo approximation provides quite reasonable agreement for a simple model using generic parameters. Longo’s concern was obtaining a model of the degradation rate of the barium dispenser cathode, and so a simple model that captured the essential features of work function variation as a function of coverage was useful. However, it does not illuminate why the work function is reduced in the first place. For that, models that address how the covering atoms create dipoles, and how those dipoles interact, are required. An oft‐used model is the Topping formula (Topping, 1927; Schmidt and Gomer, 1965). However, as it shares elements with the Gyftopoulos–Levine model, which has a good

286

KEVIN L. JENSEN

q m = 1.15 b /a = 1/2

Work function [eV]

4

a = 2.1542

Schmidt Approx

3

2

0

0.5

1.0

1.5

Coverage q FIGURE 102. The data of Schmidt (L. D. Schmidt, 1967) compared to the model of Longo et al. (1984) using generic values.

correspondence with data (even if the interpretation is a bit ambiguous; see the discussion in Haas and Thomas, 1968), its discussion is left to the literature. D. The (Modified) Gyftopoulos–Levine Model of Work Function Reduction The Gyftopoulos–Levine (GL) theory is a hard‐sphere model of the coverage atoms atop the bulk metal atoms, and it accounts for dipole and dipole‐ dipole proximity effects on the magnitude of F. It performs quite well, if one is not too persnickety in insisting that what the parameters purport to describe are physically realizable for hard spheres or whether the work function reduction is due to two sources (electronegativity differences and dipole effects) or just dipole effects. Leaving such questions of interpretation aside, the GL theory gives rather breathtaking agreement with experimental data. The account here is directed toward comparing theory to recent experimental data from photoemission studies—and to make use of small changes in atomic parameters on which the GL theory relies that have occurred in the decades since the theory made its debut. The GL theory postulates that the work function variation with coverage owes its existence to differences in electronegativity (W) and a dipole effect (d ). Electronegativity is the tendency of an atom to attract electrons to itself. Pairs of atoms have differing ability to do this, so one component of the work function represents the differing abilities of atoms to attract and retain

ELECTRON EMISSION PHYSICS

287

electrons, and the other to a dipole effect resulting from a charge redistribution. This can be written as FðyÞ ¼ W ðyÞ þ dðyÞ:

ð707Þ

Mullikan suggested that the atomic electronegativity be taken as the mean between the ionization potential and the electron affinity of an atom (Gray, 1964). The effective work function Fe of a surface was related to the electronegativity X by Gordy and Thomas (1956), who synthesized a rough formula relating the two given by Fe ½eV ¼ 2:27X ½PU þ 0:341 eV;

ð708Þ

where the work function is measured in electron volts and the electronegativity X in Pauling units. The quality of this approximation is shown in Figure 103 using values of electronegativities and work functions from the CRC tables (Weast, 1988); the approximation retains its appeal and so is adopted for historical continuity. Pauling units are such that the electronegativity of fluorine (100.45 kJ/mole ¼ 10.411 eV) is 3.98 Pauling units (PU). Therefore, a Pauling unit is 2.616 eV. The odd factor of 0.341 eV is attributed by Gyftopoulos and Levine as the energy to overcome image charge forces and therefore is the same for all metals. With the relationship between electronegativity and work function established, W(y) is taken to be the simplest polynomial that will give rise to the correct boundary conditions. These boundary conditions are as follows. For no coverage, the work function of the bulk material should arise, and the addition of a few coverage atoms should not change that. Thus, W ð0Þ ¼ fw lim dW ¼ 0 y!0 dy

ð709Þ

An analogous relationship holds for the monolayer coverage case: W ð1Þ ¼ fc lim dW ¼ 0 y!1 dy

ð710Þ

Unlike the Longo case, here fc refers to the work function of the monolayer, not the bulk material. The simplest polynomial that satisfies two boundary conditions and two derivatives at the boundaries is a cubic. It is easily shown

288

KEVIN L. JENSEN

(a) 6 Work function Gordy and Thomas f = 2.27 (x + 0.15)

Work function [eV]

5

4

3 Data from CRC handbook of chemistry and physics

2 0.5

Work function [eV]

(b)

6

1.0 1.5 2.0 Electronegativity [ pauling units]

2.5

Work function Gordy and Thomas

5

4

3

2 0

20

40 60 Atomic #

80

FIGURE 103. (a) Work function variation with electronegativity compared to the linear fit of Gordy and Thomas (1956). (b) Comparison of work function trend compared to the Gordy and Thomas model.

W ðyÞ ¼ fc þ ðfw  fc Þð1 þ 2yÞð1  yÞ2  fc þ ðfw  fc ÞHðyÞ

ð711Þ

and, as per Eqs. (377) and (378), H(0) ¼ 1 and H(1) ¼ H0 (0) ¼ H0 (1) ¼ 0, where prime indicates differentiation with respect to argument. The dipole term d(y) is more difficult. Returning to Pauling, the dipole moment between two atoms, A and B, is proportional to the difference in their electronegativities. The assumption of the GL theory is that the same

289

ELECTRON EMISSION PHYSICS

C Top

Side W

C

Perspective b

R

W FIGURE 104. Schematic of coverage atom (e.g., cesium) atop a layer of bulk (e.g., tungsten) atoms in the Gyftopoulos‐Levine model.

holds true for a site composed of four substrate atoms, represented by hard spheres, in a square array with an absorbed atom at the apex of the pyramid (Figure 104). By Eq. (708), a difference in electronegativities is tantamount to a difference in work function values apart from a constant coefficient. The distance from the center of atom ‘‘c’’ to atom ‘‘w’’ is designated R. For the four dipoles that result, only the components parallel to the vertical axis survive; the others have equal and opposite contributions. Let Mwc be the dipole moment between a c‐atom and a w‐atom. The dipole for the group of four is then Mo  4Mwc cosðbÞ, where b is the angle the line joining the atom centers makes with the vertical. GL theory suggests that Mwc is given by kðfw  fc Þ=2:27, where k ¼ 43.256 eo is a composite of factors deduced from the relationship between electronegativities and molecular dipole moments. The dipole term is then MðyÞ ¼ Mo HðyÞ Mo ¼ 4eo ro2 cosðbÞðfw  fc Þ

ð712Þ

where b and R ¼ rw þ rc are as illustrated in Figure 104. A constant radius ˚ has been introduced, and the factor parameter ro ¼ (k/2.27eo)1/2 ¼ 4.3653 A of 2.27 is the previously encountered factor relating electronegativity and

290

KEVIN L. JENSEN

work function. The cos(b) term is deduced from geometrical arguments regarding the pyramid in Figure 104 to be 2 rW 2 sin2 ðbÞ ¼ ð713Þ w R where w/(2rw)2 is the number of substrate atoms per unit area, and w the number of atoms per unit cell, where the cell size is dictated by the hard‐ sphere radius. This notation slightly departs from the path chosen by GL in terms of symbols and their meaning, but the arguments, being the same, produce the same final conclusions. Adjacent dipoles introduce a depolarizing effect such that the effective dipole moment Me(y) is the difference between the dipole moment M(y) and the depolarizing field E(y), the latter of which is proportional to Me(y) as per !3=2 9 f y Me ðyÞ ð714Þ EðyÞ ¼ 4pe0 ð2rC Þ2 where, analogous to w, the dimensionless factor f is the number of adsorbate atoms per unit cell at monolayer coverage. The effective dipole moment is found by solving Me ðyÞ ¼ MðyÞ  aEðyÞ a ¼ 4peo nr3c

ð715Þ

where a is the polarizability and the form is as given by GL. The term rc is taken as the covalent radius of the adsorbate. The factor n is slightly more tricky; it accounts for the electronic shell structure of the atom on the polarizability. Alkali metals (column 1 on the periodic table) have but one electron in the outermost shell, so n ¼ 1. Alkaline earth metals (column 2 on the periodic table) have two valence electrons, and these electrons tend to shield each other from the nucleus; thus, to account for that shielding, n ¼ 1.65 for alkaline earth elements. The dipole term d(y) is then the product of the effective dipole moment, the surface density of coating atoms, and the coverage factor, or 0 1 f Ay dðyÞ ¼ Me ðyÞ@ 2 ð2rc Þ eo 0 1 ð716Þ MðyÞ f ¼ yA 0 13=2 @ 2 eo ð2rC Þ 9a @ f A y 1þ 4pe0 ð2rC Þ2

ELECTRON EMISSION PHYSICS

291

Combining all the factors gives the work function in terms of the coverage factor y FðyÞ ¼ fC  ðfC  fW Þð1  yÞ2 ð1 þ 2yÞf1  GðyÞg 0 12 1 0 12 0 r 2 o @ A @1  @rW A A w R rC GðyÞ ¼ 0 0 13 10 1fy @1 þ n@rC A A@1 þ 9n ð f yÞ3=2 A 8 R

ð717Þ

To reiterate, the values of rw and rc are the covalent radii of the substrate and adsorbate atoms, respectively; R is the sum of them; ro is a constant radius parameter, and n depends on whether the covering is alkali or alkali earth. In Eq. (385), two parameters f and w remain to be determined by empirical data and the specifics of the system under consideration. They are not independent, as the coverage atoms reside on a surface dictated by the substrate atoms. The ratio of the substrate and adsorbate values for the number of atoms per unit cell depends on crystal face and whether the adsorbate is alkali metal or alkaline–earth metal. The nature of the surface is further dictated by which crystal plane is exposed, for example, the [100] in a body‐centered cubic (bbc) crystal. Knowledge of one crystal plane can be related to the others and therefore relates the values of f and w. Let No represent the crystal face. GL argue ½100 ) No ¼ 1 ½110 ) No ¼ 2 ½B ) No ¼ 3

ð718Þ

The first two cases appear straightforward enough, but tossing in a B demands explanation; after a certain point, the crystal face simply looks Bumpy. On a sintered tungsten surface, the best representation is to use the B value—but there are cases where crystalline surfaces are considered, and pffiffiffiffiffiffi then greater care is demanded. GL then argue that the quantities ff = N og pffiffiffiffiffiffi and fw= No g are approximately constant from one face to another. Values for a variety of coverings and substrates are given in Table 14, which updates an equivalent table in Gyftopoulos and Levine (1962). The constraint between f and w is given by the ratio of surface densities and takes the form 
 w rC 2 4 for Cs on W; Mo; Ta . . . ð719Þ ¼ 2 for Ba on Sr; Th; W . . . f rW

292

KEVIN L. JENSEN TABLE 14 COVERAGE FACTOR PARAMETERS*

Cover

rc [nm]

n

Substrate

rw [nm]

pffiffiffiffiffiffi f / No

pffiffiffiffiffiffi w/ No

Ratio

Cs Cs Cs Sr Ba Th

0.230 0.225 0.225 0.192 0.198 0.165

1 1 1 1.65 1.65 1.65

W Mo Ta W W W

0.146 0.145 0.138 0.146 0.146 0.146

0.5060 0.5161 0.4666 0.7377 0.7840 0.5440

0.8530 0.8574 0.7012 0.8530 0.8530 0.8530

4 4 4 2 2 2

*Coverage factor parameters (after Jensen, Feldman, Moody, and O’Shea, 2006a). Values of f are constructed to replicate the values of the surface densities for the adsorbate and substrate metals and other values tabulated by Gyftopoulos and Levine (1962). ‘‘Ratio’’ refers to Eq. (719). Radii are in nanometers.

Eq. (719) suggests that the ratio is an integer, but in fact, it need not be. However, it shall be treated as such and the uncertainty in the actual ratio is absorbed by defining ‘‘effective’’ values of w (or f ) such that Eq. (719) is correct. The GL theory is not without complications. Since the time of the GL article, the value of the covalent radii of various metals has changed slightly; values are taken from Winter (see WebElements; http://www.webelements.com/ webelements/). Also, there is some ambiguity surrounding how f (and hence w) is defined. Values in the literature for the surface number density of cesium on tungsten or barium on tungsten from various sources (e.g., Gyftopoulos and Levine, 1962; Haas and Thomas, 1968; Haas, Thomas, Shih, and Marrian, 1983; Schmidt, 1967; Taylor and Langmuir, 1933; Wang, 1977) tends to evolve over time. What, then, should be made of f—and more important, how should it be evaluated? The answer to such a question is intimately related to the question of how to compare different experimental data sets. E. Comparison of the Modified Gyftopoulos–Levine Model to Thermionic Data In reporting the variation of work function with coverage, the latter is often expressed in terms of fractions of a monolayer. However, this is not how coverage is measured; rather, experimental data infer ‘‘coverage’’ either by assuming a linear relation between coverage and deposition time (Wang, 1977), deposited mass measured using a quartz crystal balance (Moody et al., 2007), or other means. Therefore, experimental error, incorrect scaling factors, or both can alter the estimate of y that is quoted—and if ‘‘coverage’’ is the only parameter shown without reference to the scaling factor used, there

293

ELECTRON EMISSION PHYSICS

Work function [eV]

5 Longo-1 Longo-2 Haas Schmidt Gyfto Lev

4

3

2 0

0.2

0.4

0.6 0.8 Coverage q

1.0

1.2

FIGURE 105. Comparison of barium on tungsten as reported by several sources available in the literature using their estimates of the relationship between the experimental parameter and coverage, as compared to Gyftopoulos‐Levine theory.

is no apparent ‘‘good way’’ to compare differing measurements. That this occurs can be ascertained from comparing differing data sets from the literature for barium on tungsten; for example, see Figure 105, where a compilation of several experimental measurements is compared directly to GL theory (Longo 1 and 2 refer to Longo, Adler, and Falce, 1984; Haas refers to Haas, Shih, and Marrian, 1983; Schmidt refers to Schmidt, 1967). A similar plot can be made of, for example, cesium on tungsten. If a measure of science is reproducibility via independent measurements, then this is not reassuring. In a procedure perhaps more art than science, theoretical predictions and experimental data can be brought into line satisfactorily. A few parameters remain for which there is some ambiguity; these are the work function of the monolayer, the exact value of f, and the scale factor that must multiply a slightly off coverage estimate or which is the coefficient of the experimentally measured term (such as deposition time, etc.). The first two are tightly constrained, if they are taken to vary at all. What parameters remain to vary can then be pinned down by demanding the minimization of the least‐ squares error between GL theory and the experimental relations. When done, disparate findings coalesce satisfactorily along the GL relation for scale factors that are reasonable, as in Figure 106 for the case of barium on tungsten for a surface assumed to be bumpy (B) in deference to the polycrystalline form presumed to exist at the surface. The impact of crystal face is shown in Figure 107, which changes the value of f (that is, changing the value

294

KEVIN L. JENSEN

Work function [eV]

5 Longo-1 Longo-2 Haas Schmidt Gyfto Lev

4

3

2 0

0.2

0.4 0.6 Coverage q

0.8

1.0

FIGURE 106. A re‐analysis of the coverage factor of Figure 105 using a least‐squares analysis for the determination of the surface density parameter f.

4.5

Work function [eV]

Ba on W

3.5

B 110 100

2.5

0

0.2

0.4 0.6 Coverage q

0.8

1.0

FIGURE 107. Effects of changing the f value by considering the different crystal faces for barium on tungsten.

pffiffiffiffiffiffi of f so as to keep f = No constant) but otherwise uses the same parameters as in Figure 106. A similar result obtains for cesium on tungsten and the analysis of the data of Wang (1977) and Taylor and Langmuir (1933), as shown in Figure 108 (the Wang data used correspond to the ‘‘no‐oxygen’’ data, as oxygen tends to result in an even lower work function), again on the presumption of a bumpy surface. As with Figure 107, Figure 109 examines

295

ELECTRON EMISSION PHYSICS

4.5 Wang Taylor Gyfto Lev

Work function [eV]

4.0 3.5 3.0 2.5 2.0 1.5 0

0.2

0.4 0.6 Coverage [q ]

0.8

1.0

FIGURE 108. Same as Figure 106 but for the least‐squares analysis for determining f applied to cesium on tungsten, compared to the data of Wang (1977) and Taylor and Langmuir (1933).

4.5

Work function [eV]

Cs on W

3.5 B 110 100 2.5

1.5

0

0.2

0.4 0.6 Coverage q

0.8

1.0

FIGURE 109. Same as Figure 107 but for the parameters of cesium on tungsten.

the changes wrought by differing crystal face, but for cesium on tungsten. Overall, the experimental data are brought into remarkably consistent agreement for the values used in the Table 12. Note that the same value of f is used for Longo, Schmidt, and Haas (barium on tungsten), as well as for Taylor and Wang (cesium on tungsten). It is instructive to compare the surface number densities to values found in the literature. Gyftopoulos and Levine (1962) give the surface density of a monolayer of cesium on tungsten [purported to

296

KEVIN L. JENSEN

be due to Taylor and Langmuir (1933)] to be sCs ¼ 4:8  1014 #=cm2 , whereas Wang (1977) gives sCs ¼ 5  1014 #=cm2 . The values in Table 12 suggest sCs ¼ 4:404  1014 #=cm2 , which is in reasonable agreement. Conversely, Gyftopoulos and Levine give sBa ¼ 8:65  1014 #=cm2 for a bumpy surface [Schmidt (1967) suggests sBa ¼ 6  1014 #=cm2 for the square close‐packing density, a number between the [100] and [110] plane surface densities], whereas the table suggests s ¼ 8:78  1014 #=cm2 : again, satisfactory agreement.

F. Comparison of the Modified Gyftopoulos–Levine Model to Photoemission Data Of the variety of photocathodes that exist, the focus here is on a particular candidate intended for FELs. FELs are arguably one of the most demanding of the applications in terms of photocathode performance and characteristics, as well as the hostility of the operational environment (Colson, 2001; Neil and Merminga, 2002; O’Shea and Freund, 2001). While metal photocathodes are appreciated for their rugged behavior, their relatively low QE has always rankled. Knowledge of how coatings lowered the work function of metals in the dispenser cathode, and more importantly, how the dispenser cathode ‘‘healed’’ itself, spoke to a knowledge base that eventually found its way into speculation about how to substantially improve the QE of photocathodes. The advantage was appreciated early in the history of the FEL program (Lee and Oettinger, 1985), examined in the context of photoinjectors (Travier et al., 1995, 1997), and then various off‐the‐shelf dispenser cathodes systematically investigated (Feldman et al., 2003; Jensen, Feldman, Virgo, and O’Shea, 2003a,b; Jensen, Feldman, and O’Shea, 2004) for their utility in high‐power devices as a prelude to the development of a controlled porosity dispenser photocathode (Jensen, Feldman, Moody, and O’Shea, 2006a; Moody et al., 2007). The low work function was, as anticipated, a boon, but in an effort to characterize and baseline the impact of cesiation on metal surfaces, a systematic study was performed by Moody et al. (Moody, 2006; Moody et al., 2007) to characterize the QE as a function of surface coverage of cesium on tungsten and other metals. Those experiments became a useful testing ground for the photoemission models that have been discussed in previous sections. Here, rather than survey all such investigations, the more relevant portion focusing on the QE of cesiated surfaces as a function of wavelength will suffice. Not all cesiated surfaces for photocathodes rely on a dispenser cathode architecture—quite the contrary: for example, the present record‐holder in the pursuit of a high‐power FEL presently resides at the Thomas Jefferson

ELECTRON EMISSION PHYSICS

297

National Accelerator Facility (more commonly referred to in the FEL community as JLab) in Newport News, Virginia. Its cathode is a cesiated gallium arsenide (Cs‐GaAs) crystal (Gubeli et al., 2001). GaAs photocathodes sport high QEs of better than 10% (Neil et al., 2006; Sinclair, 2006) in addition to being a unique source of polarized electrons. The type of injector gun that uses it applies constant fields (a DC gun in the parlance) and performs bunching of the electron beam elsewhere. It has been argued, however, that if beam brightness is crucial, then RF guns (Lewellen and Brau, 2003; O’Shea, 1995) are the injector of choice, in which very high electric field gradients (on the order of 50 to 150 MV/m) rapidly accelerate short charge bunches from the photocathode (Todd, 2006). The choice of injector depends on the particular application and materials, and so great variety exists worldwide (see Colson, 2001, for a summary). One problem, however, is that photocathodes that can be used in an RF gun environment, which tends to not be as pristine as for a DC gun, are not of comparable QE. As seen from the discussion of scattering and transport to the surface, higher‐QE cathodes such as GaAs tend to have longer response times of tens of picoseconds (compared to metal photocathodes, which are essentially instantaneous; Spicer and Herrera‐Gomez, 1993), and in the generation of short bunches at the cathode, this detail can be problematic. To understand such an issue, let the incident laser pulse be Fourier transformed into a representation given by XN Il ðtÞ ¼ Io yðtÞyðT  tÞ c cosðon tÞ, ð720Þ n¼0 n where l refers to the laser wavelength, but on refers to the Fourier frequencies. If there is an emission delay time characterized by t, which has a connection to the scattering relaxation time and the depth to which the laser penetrates, then the emission current Ie(t) can be obtained from (Lewellen, 2007) Z h t  si QE t Ie ðtÞ ¼ Il ðsÞ exp  : ð721Þ t 1 t It is a straightforward problem in integration to show that Ie ðtÞ /

N X n¼0



cn 1 þ ðon tÞ

2

½ðcosðon tÞ þ on t sinðon tÞÞeT=t  1et=t ðeT=t



1Þet=t

t
The important conclusion is that all the components have tacked upon them a term of the form expðt=tÞ, indicating that the electron beam will continue to ebb out of the photocathode with an exponential tail. If the bunch lengths of the electron pulse are desired to be on the order of 10 ps

298

KEVIN L. JENSEN

(a peculiar nomenclature: the ‘‘length’’ T of a pulse is measured by its duration, usually the FWHM duration), then delay times on the order of 1 ps may be somewhat beneficial, but longer delay times destroy the pulse shape. To observe the benefit of a mild delay time, consider Figure 110; assume that the photocathode surface is at the left y‐axis. The top‐hat profile next to it is a laser pulse traveling to the left. After a time, the electron pulse (the right pulse) is moving to the right, away from the photocathode. In Figure 110a (for instantaneous emission), the electron pulse is simply the mirror image profile of the incident laser pulse. The structure on top of the pulse represents fluctuations in the laser (noise), which are a consequence of its generation by frequency‐doubling crystals and become more pronounced the shorter the wavelength of the light is made (Jensen, Feldman, Virgo, and O’Shea, 2003b), and such fluctuations are undesirable. Even a modest delay time inserted into Eq. (388) greatly curbs such fluctuations, as in Figure 110b for t ¼ 0.4 ps. Circumstances quickly degrade after that, though, and the 3.2‐ ps delay time has already morphed the electron bunch into something quite different than the top‐hat‐like laser pulse (shown in Figure 110c). The 12.8‐ps pulse of Figure 110d bears little resemblance to what was desirable in the original top‐hat distribution. Meeting the needs of pulse shaping and QE is difficult to achieve simultaneously given that metals favor the former but semiconductors the latter. A self–re‐cesiating surface based on the dispenser cathode model that is projected to have greater ruggedness in an RF environment but far higher QE than a metal photocathode has been the candidate investigated at the University of Maryland (Moody, 2006; Moody et al., 2007)—hence the interest in knowing how cesium evolves on the surface of metals. The effects on QE have been studied in a companion program at the Naval Research Laboratory (Jensen et al., 2006a). As QE can be theoretically inferred from the work function, and the work function inferred from GL theory, considering QE as a function of coverage naturally follows. Using the QE moments‐ based model and the GL work function model, such measurements can be compared to a theoretical prediction. A certain amount of latitude is offered for explorations under laboratory conditions, where the laser intensities are low and fields not appreciable. Consequently, complications that would otherwise be due to laser heating of the surface do not manifest themselves and thus temperature excursions can be neglected (though the full‐fledged temperature code is used below), as can temperature‐induced thermal desorption and migration of low–work function coatings (Gomer, 1990; Husmann, 1965; Swanson, Strayer, and Charbonnier, 1964; Taylor and Langmuir, 1933). What cannot be neglected is the quality of the surface, which can contaminate rather easily and is difficult to clean (Dowell et al., 2006; Moody et al., 2007; Sommer, 1983).

299

ELECTRON EMISSION PHYSICS

Laser/Electrons amplitude

(a) 1.2

1.0 0.8 0.6 0.4 0.2 0 −5

0

5

10

15 20 Time [ps]

25

30

35

25

30

35

25

30

35

Laser/Electrons amplitude

(b) 1.2 1.0 0.4 ps

0.8 0.6 0.4 0.2 0.0 −5

0

5

10

15 20 Time [ps]

Laser/Electrons amplitude

(c) 1.2 1.0 0.8

3.2 ps

0.6 0.4 0.2 0.0 −5

0

5

10

15 20 Time [ps]

FIGURE 110. (Continues)

300

KEVIN L. JENSEN

Laser/Electrons amplitude

(d) 1.2

1.0 12.8 ps

0.8 0.6 0.4 0.2 0.0 −5

0

5

10

15 20 Time [ps]

25

30

35

FIGURE 110. (a) Laser pulse (left structure) traveling to surface (left boundary) and the resulting electron pulse profile (right structure) traveling to the right if the delay time is 0 ps. (b) Same as (a) but for a delay time of 0.4 ps. (c) Same as (a) but for a delay time of 3.2 ps: note how even this short time degrades the top‐hat–like structure. (d) Same as (a) but for a delay time of 12.8 ps: the electron pulse bears little resemblance to the incident laser pulse.

TABLE 15 PARAMETERS FOR QUANTUM EFFICIENTY FROM A CESIATED TUNGSTEN SURFACE Parameter

Value

Unit

Field Temperature Laser intensity f Cesium atomic radius

1.7 300 0.1 1.4 0.5309

MV/m K MW/cm2 — nm 

As noted in the discussion of the thermionic cathodes, diffusion through and from pores raises questions of variable coverage. Consequently, for the comparisons here, evaporating cesium on the surface of an argon‐cleaned polycrystalline tungsten sample is used to avoid possible pooling of the cesium. Values for field, temperature, laser intensity, and other parameters for which the comparisons are made are given in Table 15. Some discussion of the parameters is needed. Although the time‐ dependent QE code is used, experimentally the pulse duration is long (on the order of tenths of seconds), which is outside the allowable bounds of the simulation model. However, the laser intensity is also very low, and since

ELECTRON EMISSION PHYSICS

301

no temperature excursion occurs, the QE of a 1‐s laser pulse is close (if not identical) to the QE of a 10‐ns pulse, and so no distinction is drawn between the two. With regard to coverage, the amount of cesium deposited is measured with a quartz crystal balance which, given the density of cesium, returns a thickness l rather than a measurement of coverage directly via l ¼ r=mA, where r is density and m and A are the mass of the deposition and the area, respectively. Given surface structure, the width of a monolayer has some ambiguity: the cesium‐to‐cesium separation distance (distance ˚ , the empirical atomic diameter of a cesium between nuclei) in bulk is 5.309 A ˚ ˚ . We atom is 5.2 A, and the cube root of the atomic volume of cesium is 4.9 A have chosen the scale factor to be the inverse cesium‐to‐cesium separation ˚ . Finally, as is evident from Figure 87, the tungsten on distance, or 5.309 A which the cesium is deposited is not pretty nor is it particularly flat. Argon cleaning of the surface roughens it. Therefore, the value of f in GL theory is not a priori unambiguous. The experimental data for QE as a function of coverage tend to show a stronger hump than GL theory with f ¼ 1. It is plausible that a rougher surface typical of sintered materials would therefore have a larger f value (as discussed by Haas and Thomas, 1968), and so the value f ¼ 1.4 was chosen. All other parameters not explicitly specified are taken from common values available in the literature (Moody, 2006, contains a complete description of the experimental arrangement and description of the surface and its treatment). The comparison between the modified GL þ time‐dependent simulation theory is shown alongside experimental data from cesium deposited onto tungsten, and the results are shown in Figure 111. Notably, even though every part of the theory is based on countless subordinate models of underlying processes and conditions, as a whole, the theory performs quite well in accounting for the qualitative and quantitative behavior of the experimental data with only one parameter ( f ) subject to some uncertainty. Similar agreement is found in comparisons of cesium on silver (Jensen et al., 2006; Moody et al., 2006). The simulation code used to generate Figure 111 can be put to other uses, as it provides a full account of the effects of temperature rise on scattering, thermal‐field emission, and photoemission in a time‐dependent framework utilizing the numerical temperature‐rise algorithms (in other words, it embodies everything that has come before). Investigating the impact of higher intensity can therefore be done theoretically. A higher temperature is correlated with a reduction in the scattering time, and therefore should correlate with a reduction in QE; alternately, an increase in applied field is correlated with a reduction in the Schottky barrier, and therefore correlated with an increase in QE. These hypotheses are tested in Figure 112 where fields of 100 MV/m and an intensity of 1 GW/cm2 are considered. The impact of a high laser intensity, even for a short 10‐ps FWHM pulse, is such that the temperature rise can climb to 895  K from a starting temperature of 300  K;

302

KEVIN L. JENSEN

Quantum efficiency [%]

(a) 0.12

375

0.08 405

532

0.04

655 808

0.00 20

0

40 60 Coverage %

(b)

100

80

100

375

10−1 Quantum efficiency [%]

80

405 532 10−2 655

10−3 808 0

20

60 40 Coverage %

(c)

405 375

Maximum QE [%]

10−1 532

655 10−2

Theory max Set A max Set B max Set C max

808 10−3 1.5

2.0

2.5

3.0

3.5

Photon energy [eV] FIGURE 111. (a) Comparison of theoretical quantum efficiency model (lines) with experimental data for cesium on tungsten at five different laser wavelengths (in nanometers). (b) Same as (a) but on log scale. (c) The maxima of the data in (a) as a function of photon energy.

303

ELECTRON EMISSION PHYSICS

the resulting degradation of QE from heightened scattering more than offsets the increase in QE from Schottky barrier lowering. Finally, it is only fitting to consider as a last example the case of cesium on copper, where the expectation is that the effects will be similarly pronounced given the good conduction characteristics of copper. In Figure 113, for standard parameters from the literature and f ¼ 1, the case of 1 GW/cm2 is compared to 1 MW/cm2 for a 10‐ps FWHM laser pulse at 355 nm, a field of

Quantum efficiency [%]

0.15

0.10

0.05 1.7 MV/m; 0.1 MW/cm^2 100 MV/m; 0.1 MW/cm^2 100 MV/m; 1.0 GW/cm^2

0.00

0

20

60 40 Coverage [%]

80 0

100

FIGURE 112. The 375‐nm line of Figure 111 for different fields and laser intensities. The former contributes to Schottky barrier lowering; the latter induces a temperature rise as the intensity increases.

0.50

Quantum efficiency [%]

Cs on Cu 0.40 0.30 0.20 1 GW/cm^2 1 MW/cm^2

0.10 0.0

0

20

40 60 Coverage [%]

80

100

FIGURE 113. Cesium on copper for differing laser intensities, in which a temperature rise in the copper is induced at the higher intensity.

304

KEVIN L. JENSEN

50 MV/m, and where the monolayer work function for cesium on copper is taken to be 1.8 eV. Copper is a better conductor of heat, so a longer pulse length of 0.1 ns (to elevate the temperature up to 1490  K) was chosen. The maximum QE of the higher‐intensity 1 GW/cm2 case is but 68% of the 1 MW/cm2 case. These cases emphasize that in the simulation of systems under extreme conditions, the various dependencies conspire in nontrivial ways to render the outcome not an a priori certainty.

V. APPENDICES A. Integrals Related to Fermi–Dirac and Bose–Einstein Statistics Integrals that appear frequently in the evaluation of energy, specific heat, and the distribution functions of fermions (s ¼ þ1) and bosons (s ¼ –1) are 0 1 ð x n1 ð 1 n1 y y @1  ð1 þn sÞAGðnÞzðnÞ  dx ¼ dx yþs yþs 2 e e 0 x 8 9 ðA1Þ < = n 1 x þ Ws ðn; xÞ ¼ ; n : ex þ s where G(n) is the gamma function, z(n) is the Riemann zeta function, and Ws is (Jensen, Feldman, Virgo, and O’Shea, 2003b) ðx yn dy: ðA2Þ Ws ðn; xÞ ¼ y y 0 ðe þ sÞð1 þ se Þ Adequate approximations to W (n,x) are therefore sought. For large x, use (Gradshteæin, Ryzhik, and Jeffrey, 1980) ð1

½lnðxÞn n! X1 sinðktÞ dx ¼ ð1Þnþk1 nþ1 , k¼1 2 sinðtÞ k 0 1 þ 2xcosðtÞ þ x

ðA3Þ

from which it can be shown   X1 Xn ð1 þ sÞ xk : n!zðnÞ  n!xn j¼1 ðsÞjþ1 ejx Ws ðn; xÞ ¼ 1  n k¼0 j k ðn  kÞ! 2 ðA4Þ

305

ELECTRON EMISSION PHYSICS

Conversely, for small x, Ws ðn; x 1Þ
2

xnþs xnþsþ2 xnþsþ4   ð3s þ 5Þðn þ sÞ 2ðs þ 7Þðn þ s þ 2Þ 24ð3s  7Þðn þ s þ 4Þ

ðA5Þ Finally, the special case s ¼ –1, n ¼ 5 is known as the Bloch–Gru¨neisen function, for which it can be shown that useful limits, accurate to four significant digits, are 0 1 8 > 1 > 2 < ðx < 0:5Þ 6x2  72 ln@1 þ x A 12 W ð5; xÞ
> > : 120zð5Þ  ðx5 þ 5x4 þ 20x3 þ 60x2 þ 120x þ 120Þex ðx > 8Þ ðA6Þ A reasonable estimate of W–(5,x) may be formed from the asymptotic limits, a ‘‘hybrid’’ polynomial (shown in Figure 69), via n o1 W ð5; xÞ
W> ð5; xÞ1 þ W< ð5; xÞ1 120zð5Þx5




80 zð5Þx2 ð18x2 þ 1Þ þ 1 3

,

ðA7Þ

which has a maximum error of 21% occurring at x ¼ 0.135. B. The Riemann Zeta Function The function z(n) is defined according to zðnÞ ¼

1 ð1  2

1 ¼ GðnÞ

1n

ð1 0

ÞGðnÞ

ð1 0

xn1 dx ex þ 1

xn1 dx ex  1

ðB1Þ

Alternately, a series definition is zðnÞ ¼ ¼

P1 1 k¼1 n k X1 ð1Þkþ1 1 k¼1 kn ð1  21n Þ

ðB2Þ

306

KEVIN L. JENSEN

Several special cases often encountered are for n ¼ 2, 3, and 4: zð2Þ ¼

p2 6

zð3Þ ¼ 1:202057 p4 zð4Þ ¼ 90

ðB3Þ

In particular, z(3) is on occasion referred to as Ape´ry’s constant. For large values of n, a convenient relation is 1 zðn þ 2Þ
ðzðnÞ þ 3Þ: 4

ðB4Þ

VI. CONCLUSION Apart from either the simple pleasure brought about by understanding why physical processes behave as they do or the slightly more complex thrill associated with using that understanding to dragoon natural phenomena to enable technological marvels, it is natural to enquire as to the utility of models of electron emission. In the discussion of the various material and operational parameters that affect emission, it has been clear that emission characteristics (by which is meant current density, emission non‐uniformity, emittance, and the other characterizations) are all affected by a host of complications. A number of them, particularly field enhancement effects due to surface structure, work function variation due to crystal faces and monolayer coatings, temperature, and other complications, have been the primary focus. Such conditions matter when thermal effects complicate field emission, dark current intrudes on photoemission, and joint thermal‐field effects are rife. The pristine and tightly constrained world of experimental characterization is then in stark contrast to either the inherent complexity of a surface or the complicated architecture of devices which exploit electron beams. Whatever utility of simple models exists therefore seems at the outset to be remarkably circumscribed. The justification for the ones considered here couples well with musings authors traditionally offer in summaries of their tomes and so such musings will be the final questions to consider: why do simple models matter, to what purposes can their improvements be directed and in what way are the models lacking or incomplete? The answers necessarily point to research underway or under consideration.

ELECTRON EMISSION PHYSICS

307

Why simple models matter is a consequence of the complexity of modern electron beam devices: modeling and simulation are often the only clairvoyant that can describe what is happening to and on account of the electron beam as it propagates, particularly as dimensions shrink in pursuit of ever higher frequency where imperfections are of greater consequence. Compression of the electron beam after it leaves the cathode region produces undesirable scalloping and halo in rf devices. High brightness sources can disrupt the electron beam by making it dependent on the variation existing at the emitter surface. The predictions are no better than the models that go into them, and such a bland observation points to the concerns here. In earlier times when the computational power brought to bear in simulating devices was far more limited, the emission models were rudimentary. As computational power increases, the impact of simulation on ‘‘first pass design success’’ is far more critical to the costly effort of designing amplifiers and rf injectors for accelerators: an account of beam evolution and spread and the impact of imperfections in the beam on the performance is essential in the design of high power devices. To accentuate the point, systems of higher operational frequency entail reduced dimensions, meaning that the quality of the electron beam in all its varied metrics has disproportionately greater impact, and the passably adequate simple models of a previous time are increasingly limited, or worse, maladapted. PIC codes such as MICHELLE are presently able to consider variation in emission over micron length scales in the modeling of macroscopic devices, so the question of variation takes on a pressing nature and is an area of active research. The more comprehensive emission models have found in the power of modern simulation codes a strong argument for their utility. The improved models are needed to address the operation of electron sources in mixed conditions when the canonical equations are inoperative, subject to conditions which vary from one regime to another, or which involve parameters that are not static throughout the emission process. Some examples suffice to convey what is envisioned.  A simple but by no means trivial complication is how much current comes from how small of an area: for field emission from sources such as Spindt cathodes, the transconductance (that is, the variation in current with applied voltage, the measure of which bears on the Class of an amplifier) depends on whether small amounts of current come from a great many points or a great deal of current from a few points; such considerations are in addition related to the scalability of the cathode (that is, whether 100 times as many emitters will produce 100 times as much current or – as is in fact more often the case – a smaller amount). A small number of emitters driven

308

KEVIN L. JENSEN

hard have a different signature and therefore impact on modulation of the resultant beam than a small amount of current per site from a great number of sites.  The nature of the surface chemistry on advanced thermionic (e.g., scandate) cathodes affects emission because it affects the mechanism (dipole versus a semiconductor model) by which the barrier to emission is lowered. In addition, the manner in which dispenser cathodes are ‘‘rejuvenated’’ by the flowing of the coating materials like barium introduces variation as a consequence of simultaneous diffusion and evaporation of the coating, producing work function variation more complex than the uniform sub‐monolayer coatings that were the focus of Gyftopoulos‐Levine theory as treated here.  For photoemitters, the aforementioned surface effects are in addition to whatever properties and their dependencies on temperature and photon frequency that exist which affect electron transport in bulk. Semiconductor photoemitters are subject to a host of complications that our focus on metals allowed us to side‐step, such as band gaps, band bending, additional scattering mechanisms, and effective mass variation. Investigations of such complications on emission are predicated on models that are more cognizant of material specific properties or behavior.  The analytical models of emittance for thermal and photoemitters is in contrast to the absence of a useful one for field emitters. In addition to the rapid decline of field away from the apex of the field emitter, the addition of a close‐proximity gate that is used to create the high fields necessary in turn significantly complicates the electron trajectories as they emerge from the gate region. The emitters are not identical, meaning that the electron trajectories are further buffeted by asymmetries in the extraction field caused by emission ‘‘hot’’ spots. In a related note, if the current density is high, then the impact of space charge in general is quite complex. Addressing such issues is the province of simulation, but given the critical nature of initial conditions, the emission theories that must be brought to bear must be more detailed than the canonical equations, or the simulations are hobbled at the outset. Finally, there is the question of the impact of additional physics that has not been considered here. A simultaneous solution of Poisson’s equation and the equations of emission are called for to investigate ‘‘quantum space charge effects’’ particularly as the dimensions of the emitters shrink into the nanoelectronics regime. A looming problem is the question of what impact nanoscale dimensions have on the emission characteristics: recall that all of the expressions for current density and emittance herein presupposed bulk‐like and nearly free electron model conditions. In the case of Spindt‐type emitters, such an approximation is perhaps adequate, but for carbon

ELECTRON EMISSION PHYSICS

309

nanotubes where the diameter of the tube is only several nanometers, assuming a bulk emission model strains credulity (apart from the question of transport between the multiwalled layers). Ab initio studies provide a means for determining what can be retained of the models considered here that will account for their Fowler‐Nordheim‐like behavior, and better models are required for when the that behavior departs from the same idyllic FN characteristics. Gratifyingly, there is much left to do. REFERENCES Abramowitz, M., and Stegun, I. A. (1965). Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables. New York: Dover Publications. Abrams, R. H., Levush, B., Mondelli, A. A., and Parker, R. K. (2001). Vacuum electronics for the 21st century. IEEE Microw. Magazine 2(3), 61–72. Adler, E., and Longo, R. (1986). Effect of nonuniform work function on space‐charge‐limited current. J. Appl. Phys. 59(4), 1022–1027. Agranat, M., Anisimov, S., and Makshantsev, B. (1988). The anomalous thermal‐radiation from metals produced by ultrashort laser‐pulses. 1. Appl. Phys. B 47(3), 209–221. Agranat, M., Anisimov, S., and Makshantsev, B. (1992). The anomalous thermal‐radiation of metals produced by ultrashort laser‐pulses. Appl. Phys. B 55(5), 451–461. Aleksandrov, A. V., Avilov, M. S., Calabrese, R., Ciullo, G., Dikansky, N. S., Guidi, V., Lamanna, P., Lenisa, P., Logachov, P. V., Novokhatsky, A. V., Tecchico, L., and Yang, B. (1995). Experimental study of the response time of GaAs as a photoemitter. Phys. Rev. E 51(2), 1449. Allison, D. K. (1981). New Eye for the Navy: The Origin of Radar A. The Naval Research Laboratory (NRL report; 8466). Washington, D.C: The Laboratory. Ancona, M. G. (1995). Thermomechanical analysis of failure of metal field emitters. J. Vac. Sci. Technol. B 13(6), 2206–2214. Anderson, D. A., Tannehill, J. C., and Pletcher, R. H. (1984). Computational Fluid Mechanics and Heat Transfer (Series in Computational Methods in Mechanics and Thermal Sciences). New York: Hemisphere Publishing. Anderson, S., Rosenzweig, J., Lesage, G., and Crane, J. (2002). Space‐charge effects in high brightness electron beam emittance measurements. Phys. Rev. Special Topics Accelerators and Beams 5(1), 014201. Ashcroft, N. W. (1966). Electron‐ion pseudopotentials in metals. Phys. Lett. 23(1), 48–50. Ashcroft, N. W., and Langreth, D. C. (1967). Compressibility and binding energy of the simple metals. Phys. Rev. 155(3), 682. Barbour, J. P., Charbonnier, F. M., Dolan, W., Dyke, W. P., Martin, E., and Trolan, J. (1960). Determination of the surface tension and surface migration constants for tungsten. Phys. Rev. 117(6), 1452–1459. Barbour, J. P., Dolan, W., Trolan, J., Martin, E., and Dyke, W. (1953). Space‐charge effects in field emission. Phys. Rev. 92(1), 45–51. Bechtel, J. H. (1975). Heating of solid targets with laser pulses. J. Appl. Phys. 46(4), 1585–1593.

310

KEVIN L. JENSEN

Bechtel, J. H., Smith, W. L., and Bloembergen, N. (1977). Two‐photon photoemission from metals induced by picosecond laser pulses. Phys. Rev. B 15(10), 4557. Bennett, H. E., and Dowell, D. H. (1999). Free‐Electron Laser Challenges II. Proceedings of SPIE. January 26–27, 1999, San Jose, California. Bellingham, W: SPIE. Berglund, C. N., and Spicer, W. E. (1964a). I. Photoemission studies of copper and silver: Experiment. Phys. Rev 136(4A), A1044. Berglund, C. N., and Spicer, W. E. (1964b). I. Photoemission studies of copper and silver: Theory. Phys. Rev. 136(4A), A1030. Binh, V. T., Purcell, S. T., Garcia, N., and Doglioni, J. (1992). Field‐emission electron spectroscopy of single‐atom tips. Phys. Rev. Lett. 69(17), 2527–2530. Blakemore, J. S. (1987). Semiconductor Statistics. New York: Dover. Bohm, D., and Hiley, B. (1985). Unbroken quantum realism, from microscopic to macroscopic levels. Phys. Rev. Lett. 55(23), 2511–2514. Bohm, D., and Staver, T. (1951). Application of collective treatment of electron and ion vibrations to theories of conductivity and superconductivity. Phys. Rev. 84(4), 836–837. Bohn, C. L., and Sideris, I. V. (2003). Fluctuations do matter: Large noise‐enhanced halos in charged‐particle beams. Phys. Rev. Lett. 91(26), 264801–1–264801–4. Brau, C. A. (1997). High‐brightness photoelectric field‐emission cathodes for free‐electron laser applications. Nucl. Instrum. Methods Phys. Res. Sect. A 393(1–3), 426–429. Brau, C. A. (1998). High‐brightness electron beams—Small free‐electron lasers. Nucl. Instrum. Methods Phys. Res. Sect. A 407(1–3), 1–7. Brennan, K. F., and Summers, C. (1987). Theory of resonant tunneling in a variably spaced multiquantum well structure: An airy function approach. J. Appl. Phys. 61, 614–623. Brodie, I. (1995). Uncertainty, topography, and work function. Phys. Rev. B 51(19), 13660–13668. Butkov, E. (1968). Mathematical Physics. Reading, MA: Addison‐Wesley. Cahay, M., Jensen, K. L., and von Allment P. (2002). Noise issue in cold cathodes for vacuum microelectronic applications. In ‘‘Noise and Fluctuations Control in Electronic Devices’’ (A. A. Balandin, ed.). Stevenson Ranch, CA: American Scientific Publishers. Campillo, I., Pitarke, J. M., Rubio, A., Zarate, E., and Echenique, P. M. (1999). Inelastic lifetimes of hot electrons in real metals. Phys. Rev. Lett. 83(11), 2230–2233. Campillo, I., Silkin, V. M., Pitarke, J. M., Chulkov, E. V., Rubio, A., and Echenique, P. M. (2000). First‐principles calculations of hot‐electron lifetimes in metals. Phys. Rev. B 61(20), 13484–13492. Carlsten, B. E., Feldman, D. W., Lumpkin, A. H., Sollid, J. E., Stein, W. E., and Warren, R. W. (1988). Emittance studies at the Los Alamos National Laboratory free electron laser. Nucl. Instrum. Methods Phys. Res. Sect. A 272(1–2), 247–256. Ceperley, D. M., and Alder, B. (1980). Ground‐state of the electron‐gas by a stochastic method. Phys. Rev. Lett 45(7), 566–569. Charbonnier, F. (1998). Arcing and voltage breakdown in vacuum microelectronics microwave devices using field emitter arrays: Causes, possible solutions, and recent progress. J. Vac. Sci. Technol. B 16(2), 880–887. Charbonnier, F. M., Southall, L. A., and Mackie, W. A. (2005). Noise and emission characteristics of Nbc/nb field emitters. J. Vac. Sci. Technol. B 23(2), 723–730. Chubun, N., and Sudakova, L. (1997). Technology and emission properties of dispenser cathode with controlled porosity. Appl. Surf. Sci. 111, 81–83. Colson, W. B. (2001). Short wavelength free electron lasers in 2000. Nucl. Instrum. Methods Phys. Res. Sect. A 475(1–3), 397–400. Corkum, P., Brunel, F., Sherman, N., and Srinivasanrao, T. (1988). Thermal response of metals to ultrashort‐pulse laser excitation. Phys. Rev. Lett. 61(25), 2886–2889.

ELECTRON EMISSION PHYSICS

311

Cortenraad, R., Denier von der Gon, A. W., Brongersma, H., Gartner, G., and Manenschijn, A. (1999). Quantitative Leis analysis of thermionic dispenser cathodes. Appl. Surf. Sci. 146(1), 69–74. Dewdney, C., and Hiley, B. (1982). A quantum potential description of one‐dimensional time‐ dependent scattering from square barriers and square wells. Found. Phys. 12(1), 27–48. Dicke, R. H., and Wittke, J. P. (1960). Introduction to Quantum Mechanics. Reading, MA: Addison‐Wesley. Dolan, W. W., and Dyke, W. P. (1954). Temperature‐and‐field emission of electrons from metals. Phys. Rev. 95(2), 327. Dowell, D. H., Davis, K. J., Friddell, K. D., Tyson, E. L., Lancaster, C. A., Milliman, L., Rodenberg, R. E., Aas, T., Bemes, M., Bethel, S. Z., Johnson, P. E., Murphy, C., Whelen, C., Busch, G. E., and Remelius, D. K. (1993). First operation of a photocathode radio frequency gun injector at high duty factor. Appl. Phys. Lett. 63(15), 2035–2037. Dowell, D. H., Joly, S., Loulergue, A., De, B., J.P, and Haouat, G. (1997). Observation of space‐ charge driven beam instabilities in a radio frequency photoinjector. Phys. of Plasmas 4(9), 3369–3379. Dowell, D. H., King, F. K., Kirby, R. E., Schmerge, J. F., and Smedley, J. M. (2006). In situ cleaning of metal cathodes using a hydrogen ion beam. Phys. Rev. Special Topics Accelerators and Beams 9(6), 063502‐1–063502‐11. DuBridge, L. A. (1933). Theory of the energy distribution of photoelectrons. Phys. Rev. 43(9), 0727–0741. Dyke, W. P., Trolan, J., Martin, E., and Barbour, J. P. (1953). The field emission initiated vacuum arc.1. Experiments on arc initiation. Phys. Rev. 91(5), 1043–1054. Esaki, L. (1973). Long Journey Into Tunneling. Nobel lecture, December 12, 1973. http:// nobelprize.org/physics/nobel_prizes/physics/laureates/1973/esaki-lecture.pdf Eyges, L. (1972). The Classical Electromagnetic Field. Reading, MA: Addison‐Wesley. Falce, L. R., and Longo, R. T. (2004). The Status of Thermionic Cathodes: Theory and Practice. Presented at the International Vacuum Electronics Conference, April 27–29, 2004. Monterey, California. Fann, W., Storz, R., Tom, H., and Bokor, J. (1992). Direct measurement of nonequilibrium electron‐energy distributions in subpicosecond laser‐heated gold‐films. Phys. Rev. Lett. 68(18), 2834–2837. Fetter, A. L., and Walecka, J. D. (1971). Quantum theory of many‐particle systems. McGraw‐Hill: San Francisco. Feldman, D. W., O’Shea, P. G., Virgo, M., and Jensen, K. L. (2003). Development of Dispenser Photocathodes for Rf Photoinjectors. Presented at the Proceedings of the IEEE Particle Accelerator Conference. Portland, Oregon. Feynman, R. P. (1987). Negative probability. In ‘‘Quantum Implications: Essays in Honour of David Bohm’’ (B. J. Hiley and F. D. Peat, eds), p. 235. New York: Routledge & Kegan Paul. Feynman, R. P. (1972). Statistical Mechanics: A Set of Lectures (Frontiers in Physics). Reading, MA: W. A. Benjamin. Forbes, R. G. (1998). Calculation of the electrical‐surface (image‐plane) position for aluminium. Ultramicroscopy 73(1–4), 31–35. Forbes, R. G. (1999a). Field emission: New theory for the derivation of emission area from a Fowler‐Nordheim plot. J. Vac. Sci. Technol. B 17(2), 526–533. Forbes, R. G. (1999b). Refining the application of Fowler‐Nordheim theory. Ultramicroscopy 79(1–4), 11–23. Forbes, R. G. (2006). Simple good approximations for the special elliptic functions in standard Fowler‐Nordheim tunneling theory for a Schottky‐Nordheim barrier. Appl. Phys. Lett. 89(11), 113122‐1–113122‐3.

312

KEVIN L. JENSEN

Forbes, R. G., and Jensen, K. L. (2001). New results in the theory of Fowler‐Nordheim plots and the modelling of hemi‐ellipsoidal emitters. Ultramicroscopy 89(1–3), 17–22. Forman, R. (1984). Correlation of electron‐emission with changes in the surface concentration of barium and oxygen on a tungsten surface. Appl. Surf. Sci. 17(4), 429–462. Fowler, R. H. (1928). The restored electron theory of metals and thermionic formulae. Proc. R. Soc. London Ser. A 117(778), 549–552. Fowler, R. H. (1931). The analysis of photoelectric sensitivity curves for clean metals. A. Various temperatures. Phys. Rev. 38(1), 45–56. Fowler, R. H., and Nordheim, L. (1928). Electron emission in intense electric fields. Proc. R. Soc. London Ser. A 119(781), 173–181. Fransen, M. J., Overwijk, M. H. F., and Kruit, P. (1999). Brightness measurements of a ZrO W Schottky electron emitter in a transmission electron microscope. Appl. Surf. Sci. 146(1–4), 357–362. Fransen, M. J., van Rooy, T. L., and Kruit, P. (1999). Field emission energy distributions from individual multiwalled carbon nanotubes. Appl. Surf. Sci. 146(1–4), 312–327. Fraser, J. S., and Sheffield, R. L. (1987). High‐brightness injectors for Rf‐driven free‐electron lasers. IEEE J. Quantum Electron. 23(9), 1489–1496. Fraser, J. S., Sheffield, R. L., Gray, E. R., and Rodenz, G. W. (1985). High‐brightness photoemitter injector for electron‐accelerators. IEEE Trans. Nucl. Sci. 32(5), 1791–1793. Frensley, W. R. (1990). Boundary conditions for open quantum systems driven far from equilibrium. Rev. Mod. Phys. 62(3), 745. Freund, H. P., and Neil, G. R. (1999). Free‐electron lasers: Vacuum electronic generators of coherent radiation. Proc. IEEE 87(5), 782–803. Fujimoto, J. G., Liu, J. M., Ippen, E. P., and Bloembergen, N. (1984). Femtosecond laser interaction with metallic tungsten and nonequilibrium electron and lattice temperatures. Phys. Rev. 53(19), 1837–1840. Fuller, G. (1974). Simple piecewise continuous approximations for Bloch‐Gruneisen functions. J Franklin I. 297(4), 293–295. Gadzuk, J. W., and Plummer, E. W. (1971). Energy distributions for thermal field emission. Phys. Rev. B 3(7), 2125. Gadzuk, J. W., and Plummer, E. W. (1973). Field emission energy distribution (feed). Rev. Mod. Phys. 45(3), 487–548. Garcia, C. H., and Brau, C. A. (2001). Electron beams formed by photoelectric field emission. Nucl. Instrum. Methods Phys. Res. Sect. A 475(1–3), 559–563. Garcia, C. H., and Brau, C. A. (2002). Pulsed photoelectric field emission from needle cathodes. Nucl. Instrum. Methods Phys. Res. Sect. A 483(1–2), 273–276. Gartner, G., Geittner, P., Raasch, D., and Wiechert, D. (1999). Supply and loss mechanisms of Ba dispenser cathodes. Appl. Surf. Sci. 146(1–4), 22–30. Gasparov, V. A., and Huguenin, R. (1993). Electron‐phonon, electron‐electron and electron surface scattering in metals from ballistic effects. Adv. Phys. 42(4), 393–521. Gilmour, A. S. (1986). Microwave Tubes (The Artech House Microwave Library). Dedham, MA: Artech House. Girardeau‐Montaut, J. P., Girardeau‐Montaut, C., Moustaizis, S. D., and Fotakis, C. (1994). Nonlinearity and inversion of femtosecond single‐ and two‐photon photoelectric emission sensitivities from gold. Appl. Phys. Lett. 26, 3664–3666. Girardeau‐Montaut, J., Afif, M., Girardeau‐Montaut, C., Moustaizis, S., and Papadogiannis, N. (1996). Aluminium electron‐phonon relaxation‐time measurement from subpicosecond nonlinear single‐photon photoelectric emission at 248 nm. Appl. Phys. A 62(1), 3–6. Girardeau‐Montaut, J., and Girardeau‐Montaut, C. (1995). Theory of ultrashort nonlinear multiphoton photoelectric‐emission from metals. Phys. Rev. B 51(19), 13560–13567.

ELECTRON EMISSION PHYSICS

313

Girardeau‐Montaut, J., Girardeau‐Montaut, C., Moustaizis, S., and Fotakis, C. (1993). High‐ current density produced by femtosecond nonlinear single‐photon photoelectric‐emission from gold. Appl. Phys. Lett. 62(4), 426–428. Girardeau‐Montaut, J., Tomas, C., and Girardeau‐Montaut, C. (1997). Dependence of photoemission efficiency on the pulsed laser cleaning of tungsten photocathodes. 2. Theory. Appl. Phys. A 64(5), 473–476. Goldstein, H. (1980). Classical Mechanics, 2nd ed. Reading, MA: Addison‐Wesley. Gomer, R. (1990). Diffusion of adsorbates on metal surfaces. Rep. Prog. Phys. 53, 917–1002. Gordy, W., and Thomas, W. J. O. (1956). Electronegativities of the elements. J. Chem. Phys. 24(2), 439–444. Gradshteæin, I. S., Ryzhik, I. M., and Jeffrey, A. (1980). Table of Integrals, Series, and Products. New York: Academic Press. Granatstein, V. L., and Armstrong, C. M. (1999). Special issue on new vistas for vacuum electronics. Proc. IEEE 87(5), 699–701. Granatstein, V. L., Parker, R. K., and Armstrong, C. M. (1999). Vacuum electronics at the dawn of the twenty‐first century. Proc. of the IEEE 87(5), 702–716. Gray, D. E. (1972). American Institute of Physics Handbook, 3rd ed. (Bruce H. Billings [and others], section editors; Dwight E. Gray, coordinating editor). New York: McGraw‐Hill. Gray, H. B. (1964). Electrons and Chemical Bonding. New York: W. A. Benjamin. Green, M. C. (1980). The M‐type cathode—No Longer Magic? Electron Devices Meeting, 1980, Vol. 26, pp. 471–474. Groning, O., Kuttel, O. M., Emmenegger, C., Groning, P., and Schlapbach, L. (2000). Field emission properties of carbon nanotubes. J. Vac. Sci. Technol. B 18(2), 665–678. Groning, O., Kuttel, O. M., Groning, P., and Schlapbach, L. (1997). Field emitted electron energy distribution from nitrogen‐containing diamondlike carbon. Appl. Phys. Lett. 71(16), 2253–2255. Groning, O., Kuttel, O. M., Groning, P., and Schlapbach, L. (1999). Field emission properties of nanocrystalline chemically vapor deposited‐diamond films. J. Vac. Sci. Technol. B 17(5), 1970–1986. Gubeli, J., Evans, R., Grippo, A., Jordan, K., Shinn, M., and Siggins, T. (2001). Jefferson Lab IR Demo FEL photocathode quantum efficiency scanner. Nucl. Instrum. Methods Phys. Res., Sect. A 475(1–3), 554–558. Gyftopoulos, E., and Levine, J. (1962). Work Function Variation of Metals Coated B. Metallic Films. J. Appl. Phys. 33(1), 67–&. Haas, G. A., Shih, A., and Marrian, C. R. K. (1983). Interatomic auger analysis of the oxidation of thin Ba films: II. Applications to impregnated cathodes. Appl. Surf. Sci. 16(1–2), 139–162. Hass, G. A., and Thomas, R. E. (1968). Thermionic emission and work function. In ‘‘Techniques of Metals Research,’’ (Bunshah R. F., ed.), Vol. VI. New York: Interscience Publishers. Haas, G. A., Thomas, R., Marrian, C., and Shih, A. (1989). Surface characterization of BaO on W. 2. Impregnated cathodes. Appl. Surf. Sci. 40(3), 277–286. Haas, G. A., Thomas, R., Shih, A., and Marrian, C. (1983). Work function measurements and their relation to modern surface‐analysis techniques. Ultramicroscopy 11(2–3), 199–206. Haas, G. A., Thomas, R., Shih, A., and Marrian, C. (1989). Surface characterization of BaO on W. 1. Deposited films. Appl. Surf. Sci. 40(3), 265–276. Hare, R. W., Hill, R. M., and Budd, C. J. (1993). Modeling charge injection and motion in solid dielectrics under high‐electric‐field. J. Appl. Phys. D 26(7), 1084–1093. Harris, J. R., Neumann, J. G., and O’Shea, P. G. (2006). Governing factors for production of photoemission‐modulated electron beams. J. Appl. Phys. 99(9), 093306. Hemstreet, L. A., and Chubb, S. R. (1993). Electronic Structure of c (2  2) Ba adsorbed on W (001). Phys. Rev. B 47(16), 10748.

314

KEVIN L. JENSEN

Hemstreet, L. A., Chubb, S. R., and Pickett, W. E. (1989). Electronic properties of stoichiometric Ba and O overlayers adsorbed on W(001). Phys. Rev. B 40(6), 3592. Herring, C., and Nichols, M. (1949). Thermionic emission. Rev. Mod. Phys. 21(2), 185–270. Hillery, M., Oconnell, R., Scully, M., and Wigner, E. (1984). Distribution‐functions in physics— fundamentals. Phys. Rep. 106(3), 121–167. Hochstadt, H. (1986). The Functions of Mathematical Physics (Dover Books on Physics and Chemistry) New York: Dover Publications. Hohenberg, P., and Kohn, W. (1964). Inhomogeneous electron gas. Phys. Rev. 136(3B), B864. Hsu, Y., and Wu, G. Y. (1992). Wigner trajectories in resonant‐tunneling diodes under transverse magnetic field. J. Appl. Phys. 71(1), 304–307. Hummel, R. E. (1992). Electronic Properties of Materials, 2nd ed. Berlin: Springer‐Verlag. Humphries, S. (1986). Principles of Charged Particle Acceleration. New York: J. Wiley. Humphries, S. (1990). Charged Particle Beams. New York: Wiley. Husmann, O. K. (1965). Alkali‐ion desorption energies on polycrystalline refractory metals at low surface coverage. Phys. Rev. 140(2A), 546–551. Ibach, H., and Lu¨th, H. (1996). Solid‐State Physics: An Introduction to Principles of Materials Science, 2nd ed. Berlin: Springer. Il’chenko, L., and Goraychuk, T. (2001). Role of the image forces potential in the formation of the potential barrier between closely spaced metals. Surf. Sci. 478(3), 169–179. Il’chenko, L., and Kryuchenko, Y. (1995). External‐field penetration effect on current‐field characteristics of metal emitters. J. Vac. Sci. Technol. B 13(2), 566–570. Imamura, S., Shiga, F., Kinoshita, K., and Suzuki, T. (1968). Double‐photon photoelectric emission from alkali antimonides. Phys. Rev. 166(2), 322–323. Ishida, H. (1990). Theory of the alkali‐metal chemisorption on metal‐surfaces. II Phys. Rev. B42(17), 10899–10911. Janak, J. F. (1969). Accurate computation of the density of states of copper. Phys. Lett. A 28(8), 570–571. Jensen, B. (1985). The Quantum Extension of the Drude–Zener Theory in Polar Semiconductors. In ‘‘Handbook of Optical Constants of Solids’’ (E. D. Palik, ed.). Orlando, FL: Academic Press. Jensen, B., and Jensen, W. D. (1991). The refractive‐index near the fundamental absorption‐edge in AlxGa1–xAs ternary compound semiconductors. IEEE J. Quantum Elect. 27(1), 40–45. Jensen, K. L. (1999). Exchange‐correlation, dipole, and image charge potentials for electron sources: Temperature and field variation of the barrier height. J. Appl. Phys. 85(5), 2667–2680. Jensen, K. L. (2000). Exchange‐correlation, dipole, and image charge potentials for electron sources: Temperature and field variation of the barrier height (Vol 85, pg 2667, 1999). J. Appl. Phys. 88(7), 4455–4456. Jensen, K. L. (2001). Theory of field mission. In ‘‘Vacuum Microelectronics’’ (W. Zhu, ed.), pp. 33–104. New York: Wiley. Jensen, K. L. (2003a). Electron emission theory and its application: Fowler–Nordheim equation and beyond. J. Vac. Sci. Technol. B 21(4), 1528–1544. Jensen, K. L. (2003b). On the application of quantum transport theory to electron sources. Ultramicroscopy 95(1–4), 29–48. Jensen, K. L. (2007). General formulation of thermal, field, and photoinduced electron emission. J. Appl. Phys. 102, 024911. Jensen, K. L., and Buot, F. A. (1989). Particle trajectory tunneling—A novel approach to quantum transport. Inst. Phys. Conf. Ser. 99, 137–140.

ELECTRON EMISSION PHYSICS

315

Jensen, K. L., and Buot, F. A. (1990). The effects of scattering on current‐voltage characteristics, transient‐response, and particle trajectories in the numerical‐simulation of resonant tunneling diodes. J. Appl. Phys. 67(12), 7602–7607. Jensen, K. L., and Buot, F. A. (1991). The methodology of simulating particle trajectories through tunneling structures using a Wigner distribution approach. IEEE Trans. Electron Dev. 38(10), 2337–2347. Jensen, K. L., and Cahay, M. (2006). General thermal‐field emission equation. Appl. Phys. Lett. 88(15), 154105. Jensen, K. L., Feldman, D. W., Moody, N. A., and O’Shea, P. G. (2006a). A photoemission model for low work function coated metal surfaces and its experimental validation. J. Appl. Phys. 99(12), 124905. Jensen, K. L., Feldman, D. W., Moody, N. A., and O’Shea, P. G. (2006b). Field‐enhanced photoemission from metals and coated materials. J. Vac. Sci. Technol. B 24(2), 863–868. Jensen, K. L., Feldman, D. W., and O’Shea, P. G. (2003). Advanced photocathode simulation and theory. Nucl. Instrum. Methods Phys. Res. Sect. A 507(1–2), 238–241. Jensen, K. L., Feldman, D. W., and O’Shea, P. G. (2004). The quantum efficiency of dispenser photocathodes: Comparison of theory to experiment. Appl. Phys. Lett. 85(22), 5448–5450. Jensen, K. L., Feldman, D. W., and O’Shea, P. G. (2005). Time dependent models of field‐ assisted photoemission. J. Vac. Sci. Technol. B 23(2), 621–631. Jensen, K. L., Feldman, D. W., Virgo, M., and O’Shea, P. G. (2003a). Infrared photoelectron emission from Scandate dispenser cathodes. Appl. Phys. Lett. 83(6), 1269–1271. Jensen, K. L., Feldman, D. W., Virgo, M., and O’Shea, P. G. (2003b). Measurement and analysis of thermal photoemission from a dispenser cathode. Physical Review Special Topics—Accelerators and Beams 6(8), 083501. Jensen, K. L., and Ganguly, A. (1993). Numerical‐simulation of field‐emission and tunneling— A comparison of the Wigner function and transmission coefficient approaches. J. Appl. Phys. 73(9), 4409–4427. Jensen, K. L., Lau, Y., and Jordan, N. (2006). Emission nonuniformity due to profilimetry variation in thermionic cathodes. Appl. Phys. Lett. 88(16), 164105. Jensen, K. L., Lau, Y., and Levush, B. (2000). Migration and escape of barium atoms in a thermionic cathode. IEEE Trans. Plasma Sci. 28(3), 772–781. Jensen, K. L., and Marrese‐Reading, C. (2003). Emission statistics and the characterization of array current. J. Vac. Sci. Technol. B 21(1), 412–417. Jensen, K. L., Moody, N. A., Feldman, D. W., O’Shea, P. G., Yater, J., and Shaw, J. (2006). The development and application of a photoemission model for cesiated photocathode surfaces. Proc. FEL 2006, Bessy, Berlin, Germany, THPPH068. Jensen, K. L., Moody, N. A., Feldman, S. W., Montgomery, E. J., and O’Shea, P. G. (2007). Photoemission from metals and cesiated surfaces. J. Appl. Phys. 102, 074902–074919. Jensen, K. L., Mukhopadhyay‐Phillips, P., Zaidman, E., Nguyen, K., Kodis, M., Malsawma, L., and Hor, C. (1997). Electron emission from a single Spindt‐type field emitter: Comparison of theory with experiment. Appl. Surf. Sci. 111, 204–212. Jensen, K. L., O’Shea, P. G., and Feldman, D. W. (2002). Generalized electron emission model for field, thermal, and photoemission. Appl. Phys. Lett. 81(20), 3867–3869. Jensen, K. L., O’Shea, P. G., Feldman, D. W., and Moody, N. A. (2006). Theoretical model of the intrinsic emittance of a photocathode. Appl. Phys. Lett. 89(22), 224103. Jensen, K. L., Phillips, P., Nguyen, K., Malsawma, L., and Hor, C. (1996). Electron emission from a single Spindt‐type field emitter structure: Correlation of theory and experiment. Appl. Phys. Lett. 68(20), 2807–2809.

316

KEVIN L. JENSEN

Jensen, K. L., and Zaidman, E. (1995). Analytic expressions for emission characteristics as a function of experimental parameters in sharp field emitter devices. J. Vac. Sci. Technol. B 13(2), 511–515. Jones, G. L., and Grant, J. T. (1983). Determination of barium and calcium evaporation rates from impregnated tungsten dispenser cathodes. Applic. Surf. Sci. 16(1–2), 25–39. Jones, W., and March, N. H. (1985). Theoretical Solid State Physics. New York: Dover Publications. Kaganov, M., Lifshitz, I., and Tanatarov, L. (1957). Relaxation between electrons and the crystalline lattice. Soviet Phys. JETP‐USSR 4(2), 173–178. Kanter, H. (1970). Slow‐electron mean free paths in aluminum, silver, and gold. Phys. Rev. B 1(2), 522–536. Kevles, D. J. (1987). The Physicists: The History of a Scientific Community in Modern America. Cambridge, MA: Harvard University Press. Kiejna, A. (1991). Image potential matched self‐consistently to an effective potential for simple‐ metal surfaces. Phys. Rev. B 43(18), 14695–14698. Kiejna, A. (1993). Surface‐properties of simple metals in a structureless pseudopotential model. Phys. Rev. B 47(12), 7361–7364. Kiejna, A. (1999). Stabilized jellium—Simple model for simple‐metal surfaces. Prog. Surf. Sci. 61(5–6), 85–125. Kiejna, A., and Wojciechowski, K. (1996). Metal Surface Electron Physics, 1st ed. New York: Elsevier Science. Kim, Y. S., and Noz, M. E. (1991). Phase Space Picture of Quantum Mechanics: Group Theoretical Approach. Teaneck, NJ: World Scientific. Kittel, C. (1962). Elementary Solid State Physics; A Short Course. New York: Wiley. Kittel, C. (1996). Introduction to Solid State Physics, 7th ed. New York: Wiley. Krolikowski, W. F., and Spicer, W. E. (1969). Photoemission studies of the noble metals. I. Copper. Phys. Rev. 185(3), 882–900. Kronig, R., and Penney, W. (1931). Quantum mechanics of electrons on crystal lattices. Proc. R. Soc. London Ser. A 130(814), 499–513. Kukkonen, C., and Wilkins, J. (1979). Electron‐electron scattering in simple metals. Phys. Rev. B 19(12), 6075–6093. Kukkonen, C. A., and Smith, H. (1973). Validity of the Born approximation as applied to electron‐electron scattering in metals: Implications for thermal conductivity. Phys. Rev. B 8(10), 4601–4606. Ladstadter, F., Hohenester, U., Puschnig, P., and Ambrosch‐Draxl, C. (2004). First‐principles calculation of hot‐electron scattering in metals. Phys. Rev. B 70(23), 235125–235110. Lang, N. D., and Kohn, W. (1973). Theory of metal surfaces—Induced surface charge and image potential. Phys. Rev. B 7(8), 3541–3550. Lang, N. D., and Kohn, W. (1970). Theory of metal surfaces: Charge density and surface energy. Phys. Rev. B 1(12), 4555–4568. Lawrence, W. E., and Wilkins, J. W. (1973). Electron‐electron scattering in the transport coefficients of simple metals. Phys. Rev. B 7(6), 2317–2332. Lee, C., and Oettinger, P. E. (1985). Current densities and closure rates in diodes containing laser‐driven, cesium‐coated thermionic cathodes. J. Appl. Phys. 58, 1996–2000. Leonard, W. F., and Martin, T. L. (1980). Electronic Structure and Transport Properties of Crystals, 1st ed. Huntington, NY: R. E. Krieger. Lerner, R. G., and Trigg, G. L. (1991). Encyclopedia of Physics, 2nd ed. New York: VCH. Levine, J., and Gyftopoulos, E. (1964a). Adsorption physics of metallic surfaces partially covered by metallic particles. I. Atom and ion desorption energies. Surf. Sci. 1(2), 171–194.

ELECTRON EMISSION PHYSICS

317

Levine, J., and Gyftopoulos, E. (1964b). Adsorption physics of metals partially covered by metallic particles. III. Equations of state and electron emission S‐curves. Surf. Sci. 1(4), 349–360. Lewellen, J. (2007). Private communication. Lewellen, J. W. (2001). Higher‐order mode Rf guns. Phys. Rev. Special Topics—Accelerators and Beams 4(4), 040101. Lewellen, J. W., and Borland, M. (2001). Emittance Measurements of the Advanced Photon Source Photocathode RF gun. Presented at the Proceedings of the 2001 Particle Accelerator Conference, 2001, Chicago, IL. Lewellen, J. W., and Brau, C. A. (2003). Rf photoelectric injectors using needle cathodes. Nucl. Instrum. Methods Phys. Res. Sect. A 507(1–2), 323–326. Lewellen, J. W., Milton, S. V., Gluskin, E., Arnold, N. D., Benson, C., Berg, W., Biedron, S. G., Borland, M., Chae, Y. C., Dejus, R. J., Den Hartog, P. K., et al. (2002). Present status and recent results from the APS SASE FEL. Nucl. Instrum. Methods Phys. Res. Sect. A 483(1–2), 40–45. Lindau, I., and Wallde´n, L. (1971). Some photoemission properties of Cu, Ag, Au and Ni. Phys. Scripta 3(2), 77–86. Logothetis, E. M., and Hartman, P. L. (1969). Laser‐induced electron emission from solids: Many‐photon photoelectric effects and thermionic emission. Phys. Rev. 187(2), 460. Longo, R. T. (1978). Long Life, High Current Density Cathodes. Technical Digest of the International Electron Devices Meeting 24, 152. Longo, R. T. (1980). A Study of Thermionic Emitters in the Regime of Practical Operation. Technical Digest of the International Electron Devices Meeting 26, 467. Longo, R. T. (2000). Dispenser Cathode Life Model: Modified to Include a Shape Factor and Refit to More Recent Life Test Data. International Vacuum Electronics Conference, 2000. Monterey, California. Longo, R. (2003). Physics of thermionic dispenser cathode aging. J. Appl. Phys. 94(10), 6966–6975. Longo, R. T., Adler, E. A., and Falce, L. R. (1984). Dispenser Cathode Life Prediction Model. Technical Digest of the International Electron Devices Meeting 30, 318. Longo, R., Tighe, W., and Harrison, C. (2002). Integrated Cathode Testing. Presented at the Vacuum Electronics Conference, 2002. IVEC 2002. Third IEEE International Vacuum Electronics Conference, 2002. Monterey, California. Lugovskoy, A. V., and Bray, I. (1998). Electron relaxation and excitation processes in metals. J. Appl. Phys. D 31(23), L78–L84. Lugovskoy, A. V., and Bray, I. (1999). Ultrafast electron dynamics in metals under laser irradiation. Phys. Rev. B 60(5), 3279–3288. Lugovskoy, A. V., and Bray, I. (2002). Electron‐electron scattering rate in thin metal films. Phys. Rev. B 65(4), 045405. Lugovskoy, A. V., Usmanov, T., and Zinoviev, A. (1994). Laser‐induced nonequilibrium phenomena on a metal surface. J. Appl. Phys. D 27(3), 628–633. Lundstrom, M. (2000). Fundamentals of Carrier Transport, 2nd ed. Cambridge, UK: Cambridge University Press. Mackie, W. A., Southall, L. A., Xie, T. B., Cabe, G. L., Charbonnier, F. M., and McClelland, P. H. (2003). Emission fluctuation and slope‐intercept plot characterization of Pt and transition metal carbide field‐emission cathodes in limited current regimes. J. Vac. Sci. Technol. B 21(4), 1574–1580. Makishima, H., Miyano, S., Imura, H., Matsuoka, J., Takemura, H., and Okamoto, A. (1999). Design and performance of traveling‐wave tubes using field emitter array cathodes. Appl. Surf. Sci. 146(1–4), 230–233.

318

KEVIN L. JENSEN

Marion, J. B., and Heald, M. A. (1980). Classical Electromagnetic Radiation, 2d ed. New York: Academic Press. Marrian, C. R. K., Haas, G. A., and Shih, A. (1983). Fast turn‐on characteristics of tungsten‐ based dispenser cathodes following gas exposures. Applic. Surf. Sci. 16(1–2), 73–92. Marrian, C. R. K., and Shih, A. (1989). The operation of coated tungsten‐based dispenser cathodes in nonideal vacuum. IEEE Trans. Electron Dev. 36(1), 173–179. Marrian, C. R. K., Shih, A., and Haas, G. A. (1983). The characterization of the surfaces of tungsten‐based dispenser cathodes. Applic. Surf. Sci. 16(1–2), 1–24. Martin, F., Garcia‐Garcia, J., Oriols, X., and Sune, J. (1999). Coupling between the Liouville equation and a classical Monte Carlo solver for the simulation of electron transport in resonant tunneling diodes. Solid State Electron. i(2), 315–323. Maruyama, T., Prepost, R., Garwin, E., Sinclair, C. K., Dunham, B., and Kalem, S. (1989). Enhanced electron‐spin polarization in photoemission from thin GaAs. Appl. Phys. Lett. 55(16), 1686–1688. Mayer, A., and Vigneron, J.‐P. (1997). Real‐space formulation of the quantum‐mechanical elastic diffusion under N‐fold axially symmetric forces. Phys. Rev. B 56(19), 12599. Mcmillan, W. L. (1968). Transition temperature of strong‐coupled superconductors. Phys. Rev. 167(2), 331–344. Michelato, P. (1997). Photocathodes for RF photoinjectors. Nucl. Instrum. Methods Phys. Res. Sect. A 393(1–3), 455–459. Millikan, R. A., and Lauritsen, C. C. (1928). Relations of field currents to thermionic currents. Proc. Natl. Acad. Sci. USA 14, 45–49. Millikan, R. A., and Lauritsen, C. C. (1929). Dependence of electron emission from metals upon field strengths and temperatures. Phys. Rev. 33(4), 0598–0604. Modinos, A. (1984). Field, Thermionic, and Secondary Electron Emission Spectroscopy. New York: Plenum Press. Mo¨nch, W. (1995). Semiconductor Surfaces and Interfaces, 2nd ed., Vol. 26, Springer Series in Surface Sciences, New York: Springer Verlag. Moody, N. A. (2006). Fabrication and Measurement of Regenerable Low Work Function Dispenser Photocathodes. PhD thesis, University of Maryland. Moody, N. A., Feldman, D. W., Jensen, K. L., Montgomery, E., Balter, A., O’Shea, P., Yater, J. E., and Shaw, J. L. (2006). Experimental progress toward low workfunction controlled porosity dispenser photocathodes. Proc. FEL 2006, Bessy, Berlin, Germany, THPPH069. Moody, N. A., Jensen, K. L., Feldman, D. W., O’Shea, P. G., and Montgomery, E. J. (2007). Prototype dispenser photocathode: Demonstration and comparison to theory. Appl. Phys. Lett. 90(11), 114108–114103. Morel, P., and Nozie´res, P. (1962). Lifetime effects in condensed helium‐3. Phys. Rev. 126(6), 1909–1918. Murphy, E. L., and Good, R. H. (1956). Thermionic emission, field emission, and the transition region. Phys. Rev. 102(6), 1464–1473. Nation, J. A., Schachter, L., Mako, F. M., Len, L. K., Peter, W., Tang, C. M., and Srinivasan‐Rao, T. (1999). Advances in cold cathode physics and technology. Proc. IEEE 87(5), 865–889. Naval Research Laboratory (NRL) Electronic Structures Database. http://cst‐www.nrl.navy. mil/ElectronicStructureDatabase/. Neil, G. R., Behre, C., Benson, S. V., Bevins, M., Biallas, G., Boyce, J., Broyce, J., Coleman, J., Dillon‐Townes, L. A., Douglas, D., Dylla, H. F., Evans, R., et al. (2006). The JLab high power ERL light source. Nucl. Instrum. Methods Phys. Res. Sect. A 557(1), 9–15.

ELECTRON EMISSION PHYSICS

319

Neil, G. R., and Merminga, L. (2002). Technical approaches for high‐average‐power free‐ electron lasers. Rev. Mod. Phys. 74(3), 685–701. Nicolaescu, D. (1993). Physical basis for applying the Fowler–Nordheim J–E relationship to experimental I–V data. J. Vac. Sci. Technol. B 11(2), 392–395. Nicolaescu, D., Filip, V., Itoh, J., and Okuyama, F. (2001). Device applied Fowler‐Nordheim relationship. Jpn. J. Appl. Phys. Part 1 40(8), 4802–4805. Nicolaescu, D., Sato, T., Nagao, M., Filip, V., Kanemaru, S., and Itoh, J. (2004). Characterization of enhanced field emission from HfC‐coated Si emitter arrays through parameter extraction. J. Vac. Sci. Technol. B 22(3), 1227–1233. O’Shea, P. G. (1995). High‐brightness Rf photocathode guns for single‐pass X‐ray free‐electron lasers. Nucl. Instrum. Methods Phys. Res., Sect. A i(1–3), 36–39. O’Shea, P. G. (1998). Reversible and irreversible emittance growth. Phys. Rev. E 57(1), 1081–1087. O’Shea, P. G., Bender, S. C., Byrd, D. A., Early, J. W., Feldman, D. W., Fortgang, C. M., Goldstein, J. C., Newnam, B. E., Sheffield, R. I., Warren, R. W., and Zaugg, T. J. (1993). Ultraviolet free‐electron laser driven by a high‐brightness 45‐MeV electron beam. Phys. Rev. Lett. 71(22), 3661–3664. O’Shea, P. G., and Bennett, H. E. (1997). Free‐Electron Laser Challenges. February 13–14, 1997, San Jose, California. Bellingham, WA: SPIE. O’Shea, P. G., and Freund, H. P. (2001). Laser technology—Free‐electron lasers: Status and applications. Science i(5523), 1853–1858. Osepchuk, J. M., and Ruden, T. E. (2005). Magnetrons. In ‘‘Encyclopedia of RF and Microwave Engineering’’ (K. Chang, ed.), Vol. 1–6, pp. 2482–2548. Hoboken, N.J.: John Wiley. Pauli, W. (1927). Zeits. f. Physik 41, 81. Papadogiannis, N., Hertz, E., Kalpouzos, C., and Charalambidis, D. (2002). Laser‐intensity effects in high‐order autocorrelation calculations. Phys. Rev. A 66(2), 025803. Papadogiannis, N., and Moustaizis, S. (2001). Ultrashort laser‐induced electron photoemission: A method to characterize metallic photocathodes. J. Appl. Phys. D 34(4), 499–505. Papadogiannis, N., Moustaizis, S., and Girardeau‐Montaut, J. (1997). Electron relaxation phenomena on a copper surface via nonlinear ultrashort single‐photon photoelectric emission. J. Appl. Phys. D i(17), 2389–2396. Papadogiannis, N., Moustaizis, S., Loukakos, P., and Kalpouzos, C. (1997). Temporal characterization of ultra short laser pulses based on multiple harmonic generation on a gold surface. Appl Phys B‐Laser O 65(3), 339–345. Parker, R. K., Abrams, R., Danly, B., and Levush, B. (2002). Vacuum electronics. IEEE Trans. Microw. Theory 50(3), 835–845. Petillo, J., Nelson, E., Deford, J., Dionne, N., and Levush, B. (2005). Recent developments to the MICHELLE 2‐D/3‐D electron gun and collector modeling dode. IEEE Trans. Electron Dev. 52(5), 742–748. Philipp, H. R., and Ehrenreich, H. (1963). Optical properties of semiconductors. Phys. Rev. 129(4), 1550–1560. Press, W. H. (1992). Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. Cambridge, UK: Cambridge University Press. Price, P. J. (1988). Theory of resonant tunneling in heterostructures. Phys. Rev. B 38(3), 1994–1998. Prutton, M. (1994). Introduction to Surface Physics. New York: Clarendon Press. Qian, Z., and Sahni, V. (2002). Quantum mechanical image potential theory. Phys. Rev. B 66 (20), 205103. Que´re´, Y. (1998). Physics of Materials. Amsterdam: Gordon and Breach Science Publishers. Quinn, J. J. (1962). Range of excited electrons in metals. Phys. Rev. 126(4), 1453–1457.

320

KEVIN L. JENSEN

Rammer, J. (2004). Quantum Transport Theory. (Frontiers in Physics; 99). Boulder, CO: Westview. Rao, T., Burrill, A., Chang, X., Smedley, J., Nishitani, T., Garcia, C. H., Poelker, M., Seddon, E., Hannon, F., Sinclair, C. K., Lewellen, J., and Feldman, D. (2006). Photocathodes for the energy recovery linacs. Nucl. Instrum. Methods Phys. Res. Sect. A 557(1), 124–130. Reichl, L. E. (1987). A Modern Course in Statistical Physics. New York: Dover. Reif, F. (1965). Fundamentals of Statistical and Thermal Physics. New York: McGraw‐Hill. Reiser, M. (1994). Theory and Design of Charged Particle Beams. (Wiley Series in Beam Physics and Accelerator Technology). New York: Wiley. Richardson, O. W., and Young, A. F. A. (1925). The thermionic work‐functions and photo‐ electric thresholds of the alkali metals. Proc. R. Soc. London Ser. A 107(743), 377–410. Ridley, B. (1977). Reconciliation of Conwell‐Weisskopf and Brooks‐Herring formulae for charged‐impurity scattering in semiconductors—Third‐body interference. J. Phys. C Solid State 10(10), 1589–1593. Ridley, B. K. (1999). Quantum Processes in Semiconductors, (4th ed.). Oxford: Clarendon Press. Riffe, D. M., Wertheim, G. K., and Citrin, P. H. (1990). Alkali metal adsorbates on W(110): Ionic, covalent, or metallic? Phys. Rev. Lett. 64, 571–574. Riffe, D., Wang, X., Downer, M., Fisher, D., Tajima, T., Erskine, J., and More, R. (1993). Femtosecond thermionic emission from metals in the space‐charge‐limited regime. J. Opt. Soc. Am. B: Opt. Phys. 10(8), 1424–1435. Rosenzweig, J., Barov, N., Hartman, S., Hogan, M., Park, S., Pellegrini, C., Travish, G., Zhang, P., David, P., Hairapetian, G., and Joshi, C. (1994). Initial measurements of the UCLA Rf photoinjector. Nucl. Instrum. Methods Phys. Res. Sect. A 341(1–3), 379–385. Schmerge, J. F., Castro, J. M., Clendenin, J. E., Colby, E. R., Dowell, D. H., Gierman, S. M., Loos, M., Nalls, M., and White, W. E. (2006). In‐situ Cleaning of Metal Photo‐cathodes in Rf Guns. Presented at Proc. FEL 2006, Bessy, Berlin, Germany. Schmidt, L. D. (1967). Adsorption of barium on tungsten: Measurements on individual crystal planes. J. Chem. Phys. 46(10), 3830–3841. Schmidt, L., and Gomer, R. (1965). Adsorption of potassium on tungsten. J. Chem. Phys. 42(10), 3573–3598. Schwoebel, P. R., Spindt, C. A., and Holland, C. (2003). Spindt cathode tip processing to enhance emission stability and high‐current performance. J. Vac. Sci. Technol. B 21(1), 433–435. Serafini, L., and Rosenzweig, J. B. (1997). Envelope analysis of intense relativistic quasilaminar beams in Rf photoinjectors: A theory of emittance compensation. Phys. Rev. E 55(6), 7565–7590. Shankar, R. (1980). Principles of Quantum Mechanics. New York: Plenum Press. Shelton, H. (1957). Thermionic emission from a planar tantalum crystal. Phys. Rev. 107(6), 1553. Shih, A., Yater, J. E., and Hor, C. (2005). Ba and BaO on W and on Sc2O3 coated W. Appl. Surf. Sci. i(1–2), 35–54. Sinclair, C. K. (2006). DC photoemission electron guns as ERL sources. Nucl. Instrum. Methods Phys. Res. Sect. A 557(1), 69–74. Sinclair, C. K. (1999). Recent Advances in Polarized Electron Sources. Presented at the Proceedings of the 1999 Particle Accelerator Conference, New York, NY. Smedley, J. M. (2001). PhD thesis. SUNY Stony Brook. Smith, A. N., Hostetler, J. L., and Norris, P. M. (1999). Nonequilibrium heating in metal films: An analytical and numerical analysis. Numer, Heat Tr. Applic. 35, 859–873. Smith, G. D. (1985). Numerical Solution of Partial Differential Equations: Finite Difference Methods (3rd ed.; Oxford Applied Mathematics and Computing Science Series). Oxford, UK: Clarendon Press.

ELECTRON EMISSION PHYSICS

321

Smith, J. R. (1969). Self‐consistent many‐electron theory of electron work functions and surface potential characteristics for selected metals. Phys. Rev. 181(2), 522–529. Smoluchowski, R. (1941). Anisotropy of the electronic work function of metals. Phys. Rev. 60(9), 661–674. Sommer, A. H. (1968). Photoemissive Materials: Preparation, Properties, and Uses. New York: Wiley. Sommer, A. H. (1983). The element of luck in research—Photocathodes 1930 to 1980. J. Vac. Sci. Tech. A 1, 119–124. Sommer, A. H., and Spicer, W. E. (1965). Photoelectric emission. In ‘‘Photoelectronic Materials and Devices’’ (S. Larach, ed.). pp. 175–221. Sommerfeld, A. (1928). Zeits. f. Physik 47, 1. Spicer, W. E. (1958). Photoemissive, photoconductive, and optical absorption studies of alkali‐ antimony compounds. Phys. Rev. 112, 114–122. Spicer, W. E. (1960). Photoemission and related properties of the alkali‐antimonides. J. Appl. Phys. 31, 2077–2084. Spicer, W. E., and Herrera‐Gomez, A. (1993). Modern theory and applications of photocathodes. Photodetectors and Power Meters Proceedings, 1993. San Diego: California. Spindt, C. A., Brodie, I., Humphrey, L., and Westerberg, E. R. (1976). Physical properties of thin‐film field emission cathodes with molybdenum cones. J. Appl. Phys. 47(12), 5248–5263. Srinivasan‐Rao, T., Fischer, J., and Tsang, T. (1991). Photoemission studies on metals using picosecond ultraviolet laser pulses. J. Appl. Phys. 69(5), 3291–3296. Srinivasanrao, T., Fischer, J., and Tsang, T. (1995). Photoemission from Mg irradiated by short‐ oulse ultraviolet and visible lasers. J. Appl. Phys. 77(3), 1275–1279. Srinivasan‐Rao, T., Schill, J., Ben Zvi, I., and Woodle, M. (1998). Sputtered magnesium as a photocathode material for Rf injectors. Rev.Sci. Instrum. 69(6), 2292–2296. Sutton, A. P. (1993). Electronic Structure of Materials. (Oxford Science Publications). Oxford: Clarendon Press. Swanson, L. W., and Strayer, R. W. (1968). Field‐electron‐microscopy studies of cesium layers on various refractory metals: Work function change. J. Chem. Phys. 48(6), 2421–2442. Swanson, L., Strayer, R. W., and Charbonnier, F. M. (1964). The effect of electric field on adsorbed layers of cesium on various refractory metals. Surf. Sci. 2, 177–187. Tallerico, P. J. (2005). Klystrons. In ‘‘Encyclopedia of RF and Microwave Engineering’’ (K. Chang, ed.), Vol. 1–6, pp. 2257–2267. Hoboken, NJ: John Wiley. Taylor, J. B., and Langmuir, I. (1933). The evaporation of atoms, ions and electrons from caesium films on tungsten. Phys. Rev. 44(6), 423–458. Tergiman, Y. S., Warda, K., Girardeau‐Montaut, C., and Girardeau‐Montaut, J. P. (1997). Metal surface photoelectric effect: Dependence on the dynamic electron distribution function. Opt. Commun. 142(1–3), 126–134. Thomas, R. E. (1985). Diffusion and desorption of barium and oxygen on tungsten dispenser cathode surfaces. Appl. Surf. Sci. 24(3–4), 538–556. Todd, A. (2006). State‐of‐the‐art electron guns and injector designs for energy recovery; inacs (ERC). Nucl. Instrum. Methods Phys. Res. Sect. A 557(1), 36–44. Tomas, C., Vinet, E., and Girardeau‐Montaut, J. (1999). Simultaneous measurements of second‐ harmonic generation and two‐photon photoelectric emission from Au. Appl. Phys. A 68(3), 315–320. Topping, J. (1927). On the mutual potential energy of a plane network of doublets. Proc. R. Soc. London Ser. A 114(766), 67–72. Travier, C. (1991). Rf guns—Bright injectors for FEL. Nucl. Instrum. Methods Phys. Res. Sect. A 304(1–3), 285–296.

322

KEVIN L. JENSEN

Travier, C. (1994). An introduction to photo‐injector design. Nucl. Instrum. Methods Phys. Res. Sect. A 340(1), 26–39. Travier, C., Devanz, G., Leblond, B., and Mouton, B. (1997). Experimental characterization of Candela photo‐injector. Nucl. Instrum. Methods Phys. Res. Sect. A 393(1–3), 451–454. Travier, C., Leblond, B., Bernard, M., Cayla, J., Thomas, P., and Georges, P. (1995). Candela photo‐injector experimental results with a dispenser photocathode. Proc. 1995 Particle Accelerator Conference 2, 945–947. Tsang, T., Srinivasanrao, T., and Fischer, J. (1992). Photoemission using femtosecond laser‐ pulses. Nucl. Instrum. Methods Phys. Res. Sect. A 318(1–3), 270–274. Tsu, R., and Esaki, L. (1973). Tunneling in a finite superlattice. Appl. Phys. Lett. 22(11), 562–564. Vancil, B., and Wintucky, E. G. (2006). Reservoir cathodes—Recent development. Presented at the Technical Digest of the IEEE Vacuum Electronics Conference, 2006, held jointly with 2006 IEEE International Vacuum Electron Sources. Vassell, M. D., Lee, J., and Lockwood, H. F. (1983). Multibarrier tunneling in Gal‐Xalxas/Gaas heterostructures. J. Appl. Phys. 54, 5206–5213. Vigier, J.‐P., Dewdney, C., Holland, P. R., and Kyprianidis, A. (1987). Causal particle trajectories and the interpretation of quantum mechanics. In ‘‘In Quantum implications: essays in honour of David Bohm’’ (B. J. Hiley and F. D. Peat, eds), pp. 169–204. Wagner, D. K., and Bowers, R. (1978). Radio‐frequency size effect—Tool for investigation of conduction electron‐scattering in metals. Adv. Phys. 27(5), 651–746. Wang, C. (1977). High photoemission efficiency of submonolayer cesium‐covered surfaces. J. Appl. Phys. 48(4), 1477–1479. Watson, G. N. (1995). A Treatise on the Theory pf Bessel Functions (2nd ed. Cambridge Mathematical Library). Cambridge, Cambridge University Press. Weast, R. C. (1988). CRC Handbook of Chemistry and Physics, 1st student ed. Boca Raton, FL: CRC Press. Wertheim, G. K., Riffe, D. M., Smith, N. V., and Citrin, P. H. (1992). Electron mean free paths in the alkali‐metals. Phys. Rev. B 46(4), 1955–1959. Westlake, D., and Alfred, L. (1968). Determination of Debye characteristic temperature of vanadium from Bloch‐Gruneisen relation. J. Phys. Chem. Solids 29(11), 1931–1934. Whaley, D., Gannon, B., Smith, C., Armstrong, C., and Spindt, C. (2000). Application of field emitter arrays to microwave power amplifiers. IEEE Trans. Plasma Sci. 28(3), 727–747. White, G. K., and Woods, S. B. (1959). Electrical and thermal resistivity of the transition elements at low temperatures. Philos. Trans. R. Soc. London Ser. A 251(995), 273–302. Wilson, A. H. (1938). The electrical conductivity of the transition metals. Proc. R. Soc. London Ser. A 167(A931), 0580–0593. Winter, M. WebElements (1993). Http://www.webelements.com/webelements/ http://www. webelements.com. Wright, O., and Gusev, V. (1995). Ultrafast feneration of acoustic‐waves in copper. IEEE Trans. Ultrason. Ferroelectr. Frequency Control 42(3), 331–338. Xu, N., Chen, J., and Deng, S. (2000). Physical origin of nonlinearity in the Fowler‐Nordheim plot of field‐induced emission from amorphous diamond films: Thermionic emission to field emission. Appl. Phys. Lett. 76(17), 2463–2465. Yamamoto, S. (2006). Fundamental physics of vacuum electron sources. Rep. Prog. Phys. 69(1), 181–232. Yilbas, B., and Arif, A. (2006). Laser short pulse heating: Influence of pulse intensity on temperature and stress fields. Appl. Surf. Sci. 252(24), 8428–8437. Yilbas, B. (2006). Improved formulation of electron kinetic theory approach for laser ultra‐ short‐pulse heating. Int. J. Heat Mass Tran. 49(13–14), 2227–2238.

ELECTRON EMISSION PHYSICS

323

Young, R. D. (1959). Theoretical total‐energy distribution of field‐emitted electrons. Phys. Rev. 113(1), 110–114. Young, R. D., and Mu¨ller, E. W. (1959). Experimental measurement of the total‐energy distribution of field‐emitted electrons. Phys. Rev. 113(1), 115–120. Zhou, F., Ben‐Zvi, I., Babzien, M., Chang, X. Y., Doyuran, A., Malone, R., Wang, X. J., and Yakimenko, V. (2002). Experimental characterization of emittance growth induced by the nonuniform transverse laser distribution in a photoinjector. Phys. Rev. Special Top.— Accelerators and Beams 5(9), 094203. Zhu, W. (2001). Vacuum Microelectronics. New York: Wiley. Zhukov, V., Chulkov, E., and Echenique, P. (2006). Lifetimes and inelastic mean free path of low‐energy excited electrons in Fe, Ni, Pt, and Au: Ab Initio GWþT calculations. Phys. Rev. B 73(12), 125105. Ziman, J. M. (1985). Principles of the theory of solids, 2nd ed. Cambridge: Academic Press. Ziman, J. M. (2001). Electrons and Phonons: The Theory of Transport Phenomena in Solids. (Oxford Classic Texts in the Physical Sciences). Oxford: Oxford University Press. Zuber, J., Jensen, K. L., and Sullivan, T. (2002). A. Analytical solution for microtip field emission current and effective emission area. J. Appl. Phys. 91(11), 9379–9384.

This page intentionally left blank

Index

random-phase, 42, 179 Spindt quadratic, 116 Stirling’s, 132 Thomas-Fermi, 42 APS. See Advanced photon source Area under the curve (AUC), 65 Area-under-the-potential, 69 Argon-cleaned polycrystalline tungsten, 300 Asymptotic expressions, 109, 232 Asymptotic limits, 109, 174 Atomic polarizability, 157, 162 Atoms chain to lattice transition of, 191 coverage, 289 monatomic linear chain of, 186–194 multielectron, 13 polarized, 281 radial hydrogen, 13 AUC. See Area under the curve Awkward asymptotic expressions, 109

A Absorption probability, 267 Accelerator community, 263 Acoustic phonon, 193 scattering, 208 time, 212 Ad hoc simulations, 118 Adjacent dipoles, 290 Adjacent pulses, 231 Advanced photon source (APS), 270 After conditioning, 130 Airy coeYcients, 77 Airy functions, 74 approach, 71–80, 101 polynomial, 75, 76 Wronskians of, 80 Alkaline-earth metals, 290, 291 Analytical formula, 279 Angstrom-scale distances, 38 Angular integrations, 183, 205, 221 Annihilation operators, 25, 200 Anticommutator, 51 Apex configuration, 129 Approximation, 179 born, 33, 175 crue quadratic, 90 emission equation, 106–110 finite diVerence, 241 Forbes, 137 Fowler-Dubridge, 257 Friedel, 44 harmonic oscillator, 191 hyperbolic tangent, 41, 44, 94 image charge, 40–46 local-density, 31 Longo, 285 polynomial, 137 pseudopotential, 32

B Balance detailed, 181 quartz crystal, 292 Band bending, 20–22 Band structure, 13–20 band bending, 20–22 semiconductors, 20 Bare metals, 148–150 density of states of, with respect to nearly free electron gas model, 264 quantum eYciency of, 260–273 Barium, 283, 293 325

326 Barrier exact quadratic, transmission probability, 86 Gaussian potential, 61–62 height, 276 image charge, 87–94 k value, 17 maximum, 83, 91–94 multiple square, 69–71 quadratic, 85–86, 90–91 Schottky, lowering, 222, 303 single, 70 square, 67–69 surface, 22–40 triangular, 80–85 BE. See Bose-Einstein Bechtel-like conditions, 253 Behavior Bloch-Grneisen, 207 of electron-electron relaxation time, 185 low-temperature, 213 Bessel functions, 33 Beta dactor, 119 BH. See Brooks-Herring Bloch-Grneisen behavior, 207 Bloch-Grneisen function, 208, 210 Blue line results, 261 Bohm approach, 62–64 Bohm trajectories, 57 Bohm-Staver relation, 195, 198, 208 Bohm-Staver result, 197 Boltzmann’s constant, 6 Boltzmann’s equation, 158 Boltzmann’s transport equation (BTE), 11, 53 Born approximation, 32, 175 Bose-Einstein (BE), 7 Bose-Einstein statistics, 304–305 Bra-ket notation, 17–18 Brooks-Herring (BH), 179 BTE. See Boltzmann’s transport equation

INDEX

C Calculated peak temperature, 225 Canonical copper, 232 Canyonlike complexity, 266 Carbon nanotubes, 120 Cathodes. See also Photocathodes dispenser, 283, 284, 296 lower-work function, 282 Schottky emission, 105 sintered tungsten dispenser, 262 CDS. See Central diVerence schemes Central diVerence scheme (CDS), 241 Cesium, 293, 296 Charged impurity relaxation time, 177–179 Chemical potential, 6, 9–11, 92 Classical distribution function approach, 47–49 Classical image charge, 5 Classically forbidden region, 34 Clean tungsten emitter, 119 CoeYcients airy, 77 matrix, 244 requisite, 254 temperature-dependent, 247–253 Collision integral, 200 Collision operator, 180 Complex conjugation, 203 Complexity canyonlike, 266 considerable, 146 Conductivity, 165–174 DC, 167 electrical, 165–167 photoemission, 165–174 thermal, 167–170 thermal, data, 214 Conjugation, 203 Considerable complexity, 146 Constants Boltzmann’s, 6 dielectric, 15, 154–156 electron-phonon coupling, 236

327

INDEX

fundamental, 4 potential segment, 65–67 representations, 138 Richardson, 103 Contamination, 269–273 Continuity equation, 49 Contour map, 17 Copper canonical, 232 parameters, 93, 131, 233, 238 reflectivity of, 266 Core radius, 39 Correlation energy, 29 Coulomb potential, 13, 195 Coupled heat equations, 234–235 Coupled temperature equations, 248 Coupled thermal equations numerical solution of, 239–253 nature of problem, 239–240 ordinary diVerential equations, 240–246 numerically solving, 247–253 Coverage atom, 289 Coverage factor parameters, 239 Crank-Nicolson method, 245 Creation/annihilation operators, 27, 58 Crue quadratic approximations, 90 Crystal faces diVerent, 295 work function, 261–263 Current of energy, 168 Current of heat, 168

D D electrons, 217 D’Alambertian operator, 198 DC. See Direct current DC conductivity, 167 Debye cutoV, 173 Debye frequency, 191, 206 Debye temperature, 208 Deformation potential, 199, 218 Delta-function-like pulse, 227

Denominator, 221 Detailed balance, 181 Diatomic case, 186 Dielectric constant, 15, 154–156 DiVerent crystal faces, 295 Dimensionless parameter, 28 Dipoles adjacent, 290 contribution, 37 eVective, moment, 289 eVects, 33–40, 286 molecular, moments, 289 term, 44 Dirac delta function, 14, 101, 169, 176, 205 Direct current (DC), 160 Direct numerical evaluation, 112 Discrete representation, 97 Dispenser cathodes, 283, 284, 296 Distribution classical, function approach, 47–49 electron, 214 emitted, 126 energy, 127 FD, 27 function, 48, 257 Gaussian, 48, 56 general, 48 normal emission, 126 theoretical energy, 127 Wigner Distribution Function Approach, 52–62 Downwind diVerence schemes, 242 Drive lasers, 222–223, 250 Drude model, 156–162 Drude relations, 160

E Edison eVect, 279 EVective dipole moment, 290 EVective emission area, 269–273 EVective quantum eYciency, 273 EYciency, 273. See also Quantum eYciency

328 Electrical conductivity, 165–167 Electrochemical potential, 166 Electromagnetic wave, 155 Electron(s) background contribution, 24 collisions, 236 D, 217 density, 36 distribution, 214 electron collisions, 184 electron relaxation time, 185 behavior of, 185 electron scattering, 180–185, 220, 221 emission, 22–40 eVects, 261 physics, 279 surface eVects/origins of work function, 22–31 energies, 210 fast electron-electron scattering mechanism, 236 FEL, 147 free electron gas, 5–11, 264 free, gas model, 264 free, model, 8 gas, 197, 238 high-power free, lasers, 222 IEDM, 282 multi, 13 nearly, electron gas, 11–22 number density, 237 photoemitted, 221 photoexcited, 219, 236, 278 uniform, density, 31 zero-temperature, gas, 151 Electronegativity, 286, 288, 289 Electron-phonon coupling constant, 236 coupling factor, 236–239 relaxation time, 238 scattering, 194–212 calculations, 191 Elliptical integrals, 131–136 Emission. See also Field emission; Photoemission eVective area, 270–274

INDEX

electron, 22–40 enhanced, 279–304 equation integrals, 106–110 general, equation, 105 low field thermionic, studies, 103 normal, distribution, 126 in thermal-field transition region revisited, 136–139 thermionic, 279 triangular barrier, probability, 85 Emittance, 273, 277 Emitted distribution, 126 Emitter clean tungsten, 119 field, 265 microfabricated, 270 Spindt-type, 128 thermionic, 270 tungsten, 119 Energy correlation, 29 current of, 168 distribution, 127 exchange/correlation, 30 Gaussian, analyzer, 126 high, photons, 215 kinetic, 27, 34, 188, 219 LEUTL, 270 photoexcited electron, 278 photon, 303 Rydberg, 29, 185 stupidity, 29 theoretical, distribution, 127 total, per unit volume, 29 Enhanced emission, 279–304 less simple model of, 282–286 simple model of, 281–282 Equations approximation, emission, 106–110 Boltzmann, 158 Boltzmann transport, 11, 53 continuity, 49 coupled heat, 234–235 coupled temperature, 248 coupled thermal, 239–253

INDEX

Fowler-Nordheim, 104–106, 143 Fowler-Richardson-LaueDushmann, 117 general emission, 105 general thermal-field, 131, 139 linearized Boltzmann, 165 ordinary diVerential, 240–246 parabolic, 240 Poisson’s, 14, 45, 195 revised FN-RLD, 118–130 Richardson, 166 RLD, 104–106, 110–118 Schro¨dinger’s, 12, 15–16, 162 thermal-field, 139–146 Equivalent formulations, 21 Error function, 228 Escape cone, 151–154 Exact quadratic barrier transmission probability, 86 Exactly solvable models, 65–85 airy functions approach, 71–80 multiple square barriers, 69–71 square barrier, 67–69 triangular barrier, 80–85 wave function methodology for constant potential segment, 65–67 Exchange term, 27 Exchange/correlation energy, 30 Exchange-correlation potential, 31 Experimental reflectance, 163 Experimental relations, 213

F Fast electron-electron scattering mechanism, 236 FD. See Fermi-Dirac; FowlerDubridge formulation FEL. See Free electron laser Femtoseconds, 206 Fermi level, 138, 168, 220, 237, 255 Fermi momentum, 10 hkF, 8, 182 Fermi-Dirac (FD), 7 distribution function, 27

329 integral, 9–10 integral circles, 9 integrals related to, 304–305 Fermi’s golden rule, 174–177 Feynman diagrams, 29, 180 Fick’s law, 225 Field emission, 47–146, 273 current density, 47–64 in Bohm approach, 62–64 in classical distribution function approach, 47–49 in Gaussian potential barrier, 61–62 in Schro¨dinger/Heisenberg representations, 47, 49–53 in Wigner distribution function approach, 52–62 exactly solvable models of, 65–85 numerical methods of, 94–101 numerical treatment of image charge potential, 95–99 numerical treatment of quadratic potential, 95 resonant tunneling, 99–102 recent revisions of standard thermal/field models in, 131–139 emission in thermal-field transition region revisited, 136–139 Forbes approach to evaluation of elliptical integrals, 131–136 revised FN-RLD equation/ inference of work function from experimental data in, 118–130 mixed thermal-field conditions, 123–126 slope-intercept methods applied to field emission, 127–130 thermionic emission, 121–123 thermal/equation, 102–118 emission equation integrals/their approximation, 106–110 Field emitter, 265 Field enhancement, 264–269

330 Field operator notation, 23 Field-dependent area factor, 120 Field/thermionic emission fundamentals, 4–46 free electron gas, 5–11 image charge approximation, 40–46 nearly free electron gas, 11–22 surface barriers, 22–40 unit note, 4–5 Finite diVerence approximation, 241 FN. See Fowler-Nordheim Forbes approach, for elliptical integrals, 131–136 Forbes approximation, 137 Forbes expression, 140 Force-free evolution, 56 Formula analytical, 279 asymptotic limit, 109 Fourier components, 175 Fourier transform, 157 Fowler-Dubridge approximation, 257 Fowler-Dubridge formulation (FD), 257 Fowler-Dubridge function, 107, 152, 220, 265 comparison of, 153, 154 Fowler-Dubridge model, 150, 219, 222, 269 revisions to modified, 253–255 Fowler-Dubridge probability ratio, 259 Fowler-Nordheim (FN), 45 Fowler-Nordheim equations, 104–106, 143 Fowler-Nordheim factors, 131 Fowler-Richardson-Laue-Dushmann equation, 117 Fractional monolayer coverage, 285 Free electron gas, 5–11 chemical potential of, 9–11 density of states with respect to nearly, 264 quantum statistical mechanics of, 5–8

INDEX

Free electron laser (FEL), 147 Free electron model, 8 Frequencies, 191 Debye, 191, 206 optical, 162 resonance, 162–164 Friedel approximation, 44 Friedel oscillations, 36, 37, 42 Full-width-at-half-max (FWHM), 115 Functions airy, 71–80, 74, 75, 76, 101 Bessel, 33 Bloch-Grneisen, 208, 210 classical distribution, approach, 47–49 crystal faces, 261–263 Delta-function-like pulse, 227 Dirac delta, 14, 101, 169, 176, 205 distribution, 48, 257 Drude model, 156–162 error, 228 Fermi-Dirac distribution, 27 Fowler-Dubridge, 107, 152, 153, 154, 220, 265 general distribution, 48 ground-state Wigner, 59–60 Heaviside step, 27, 33, 151 Kronecker delta, 18 Riemann zeta, 9, 108, 152, 305–306 Wigner, 158 Wigner distribution, 52–62, 61 Fundamental constants, 4 FWHM. See Full-width-at-half-max

G Gas electron, 197, 238 free electron, 5–11 nearly free electron, 11–22 zero-temperature electron, 151 Gaussian distribution, 48 Gaussian distribution density, 56 Gaussian energy analyzer, 126

331

INDEX

Gaussian laser pulse, 253 Gaussian potential barrier, 61–62 General distribution, 48 General emission equation, 105 General thermal-field equation, 131, 139 Gold parameters, 238 Ground-state Wigner function, 59–60 Gyftopoulos-Levine model, 286–292 photoemission compared to, 296–304 thermionic comparison to, 292–296 Gyftopoulos-Levine theory, 13, 40, 283

H Harmonic number, 223 Harmonic oscillator, 57–62, 187 Harmonic oscillator approximation, 191 Heat of solids, 171–174 Heat/corresponding temperature, 225–227 Heaviside step function, 27, 33, 151 Heisenberg picture, 50 Heisenberg representations, 49–52 Hermite polynomials, 60 High electric field gradients, 297 High-energy photons, 215 Higher-order derivatives, 241 High-power free-electron lasers, 222 High-temperature representation, 207 Hyperbolic tangent approximation, 41, 44, 94 Hyperellipsoid, 143

I IEDM. See International Electron Devices Meeting Image charge approximation, 40–46 analytical image charge potential of, 43–46 classical treatment of, 40–42

expansion  near E ¼  of, 88–90 expansion  near E ¼ : quadratic barrier of, 90–91 quantum mechanical treatment of, 42–43 reflection above barrier maximum in, 91–94 Image charge barrier, 87–94 Image charge potential, 95–99 Image pulse, 227 Implicit schemes, 247 Incidence angle, 268 Increased absorption, 267 index of refraction/ reflectivity, 154–156 Inference of work function from experimental data, 118–130 Integrand components, 253 Integrations angular, 183, 205, 221 time, 231 International Electron Devices Meeting (IEDM), 282 Intrinsic emittance, 145 Inversion invariant, 203 Ion core eVects, 31–33 Ion-electric cloud, 156 Ionized scattering site, 177 Isotropic crystal, 172 Isotropic system, 188

J JeVreys-Wenzel-Kramers-Brillouin (JWKB), 62 Jellium, 13 JWKB. See JeVreys-Wenzel-KramersBrillouin

K Kinetic energy, 27, 188 component, 219 operator, 23 Kronecker delta function, 18 Kronig-Penney model, 13–20, 31, 47

332

L Large argument case, 78–79 Laser(s) drive, 222–223, 250 free electron, 147 heating, 223 high-power free-electron, 222 intensities, 304 intensity, 232 pulse, 252, 300 temperature of, illuminated surface, 222–239 Laser pulse Gaussian, 253 maximum, 249 simple model of temperature increase due to, 223–225 Lattice temperature tracks, 250 LDA. See Local-density approximation Least-squares analysis, 295 LEUTL. See Low-energy undulator test line LINAC. See Linear accelerator Linear accelerator (LINAC), 144 Linear beam, 281 Linear segment potential, 100 Linearized Boltzmann equation, 165 Liouville’s theorem, 144 Liquid nitrogen temperature, 218 Local-density approximation (LDA), 31 Longo approximation, 285 Lorentzian components, 163 Low field thermionic emission studies, 103 Low-energy undulator test line (LEUTL), 270 Lower-work function cathode, 282 Low-temperature behavior, 213 Low-temperature leading-order limit, 221

INDEX

Low-work function coatings, 279–304 less simple model of, 282–286 simple model of, 281–282

M Macroscopic fields, 163, 226 Macroscopic surface, 268 Macroscopic viewpoint, 226 Magnetron-sputtered lead, 267 Matrix notation, 16 Matthiessen’s rule, 212–215 Maximum temperature, 229 Maxwell-Boltzmann statistics, 6–7 Metallic-like parameters, 255 Metals alkaline-earth, 290, 291 bare, 148–150, 261–274 photocathodes, 150, 260 Meter-kilogram-second-ampere (MKSA), 4 Microcrystalline surfaces, 270 Microfabricated emitter arrays, 270 Micron-scale resolution, 265 Microscopic scale, 265 Millikan’s erroneous conjecture, 106 MKSA. See Meter-kilogramsecond-ampere Molecular dipole moments, 289 Moments-based approach, 215, 255–261 Momentum, 177 delta function, 200 eigenstate, 51 like variables, 33 relaxation time, 206, 211 Monatomic case, 186 Monatomic linear chain of atoms, 186–194 Monatomic system, 188 Monolayer coverage, 283, 287 Monte Carlo simulations, 216 Multielectron atoms, 13 Multiple pulses, 227–233

333

INDEX

Multiple reflections, 264–269 Multiple square barriers, 69–71

N NANO units, 5 Nanotubes, 120 National Accelerator Facility, 296 Naval Research Laboratory (NRL), 279, 298 NEA. See Negative Electron AYnity Nearly free electron gas, 11–22 band structure/Kronig-Penney model and, 13–22 hydrogen atom and, 11–12 Negative Electron AYnity (NEA), 148 Nondegenerate, 20 Nontrivial matrix, 245 Normal emission distribution, 126 Normal incidence, 277 Normalization, 59 Normalized brightness, 145 Nottingham heating, 113 NRL. See Naval Research Laboratory Numerical treatment of quadratic potential, 95 Numerically evaluated transmission probability, 84

O Ohm’s law, 160 Operators annihilation, 25, 200 collision, 180 creation/annihilation, 27, 58 D’Alambertian, 198 field, notation, 23 formalism, 202 kinetic energy, 24 Optical frequencies, 162 Optical phonons, 193 Orbitals, 12 Order unity, 185 Order zero, 33

Ordinary diVerential equations, 240–246 Oscillators harmonic, 57–62, 191 strength term, 162

P Parabolic equations, 240 Parameters copper, 238 copper-like, 93, 131, 233 coverage factor, 239 dimensionless, 28 gold, 238 Thomas-Fermi, 195 tungsten, 223, 293 Particular finite duration, 227 Pauling radius, 39 Pauling units, 287 Penetration depth, 154–164 dielectric constant/index of refraction/reflectivity and, 154–156 Drude model and, 156–162 quantum extension/resonance frequencies, 162–164 Periodic permanent magnet focusing, 281 Phase space description, 11 Phonons acoustic, 193, 208, 212 interaction terms, 201 optical, 193 relaxation time, 212 Photocathodes, 147, 222–223, 225 drive laser combinations, 250 emittance/brightness of, 274–279 metal, 150, 260 simulation algorithm, 232 surface, 278 Photoemission, 147–279 background of, 147–148 conductivity, 165–174 electrical conductivity, 165–167 heat of solids, 171–174

334 Photoemission (Cont.) thermal conductivity, 167–170 Wiedemann-Franz Law, 170–171 emittance/brightness of photocathodes and, 274–279 modified Gyftopoulos-Levine model compared to, 296–304 numerical solution of coupled thermal equations of, 239–253 probability of, 151–154 quantum eYciency of bare metals and, 148–150, 260–273 quantum eYciency revisited/ moments-based approach and, 255–259 reflection/penetration depth of, 154–164 revision to modified FowlerDubridge model and, 253–255 scattering factor of, 215–222 scattering rates of, 174–215 temperature of laser-illuminated surface of, 222–239 wavelengths of, 212 Photoemitted electrons, 221 Photoexcitation, 214 Photoexcited electrons, 219, 236 energy, 278 Photons energy, 303 high-energy, 215 Poisson’s equation, 14, 45, 195 Polarization diagrams, 29 Polarized atom, 281 Polycrystalline form, 293 Polynomial airy functions, 75, 76 Polynomial approximation, 137 Pore-to-pore separation, 283 Post-conditioning current-voltage plots, 129 Potassium, 283 Power tubes, 279 Predictor-corrector methods, 246 Predictor-corrector schemes, 247 Probability absorption, 267

INDEX

density, 13 Fowler-Dubridge, 259 numerically evaluated transmission, 84 of photoemission, 151–154 single barrier transmission, 70 transmission, 52, 68, 69, 101 Proper method, 99 Pseudopotential approximation, 32 Pulses adjacent, 231 delta-function-like, 227 Gaussian laser, 253 image, 227 laser, 223–225, 252, 300 laser, maximum, 249 multiple, 227–233 Pyramidal depressions, 40

Q QE. See Quantum eYciency Quadratic barrier, 85–86 Quadratic method, 99 Quadratic potential, 93, 95 Quantum eYciency (QE), 148–150, 222, 263 of bare metals, 260–273 contamination/eVective emission area, 269–273 density of states with respect to nearly free electron gas model, 264 surface structure/multiple reflections/field enhancement, 264–269 variation of work function with crystal face, 261–263 plots, 272 revisited/moments-based approach, 255–259 Quantum extension, 162–164 Quantum mechanical treatment, 42–43 Quantum potential, 63 Quantum statistical mechanics, 5–8

INDEX

Quantum trajectory, 57 Quartz crystal balance, 292

R Radial hydrogen atom, 13 Random-phase approximation (RPA), 42, 179 Reflections, 154–164 above barrier maximum, 91–94 multiple, 264–269 Relaxation time, 165, 176 charged impurity, 177–179 phonons, 212 Requisite coeYcients, 254 Residual resistivity, 212 Resistivity values, 211 Resonance frequencies, 162–164 Resonant tunneling, 99–102 Resonant tunneling diode (RTD), 63 Revised FN, 110–118 Revised FN-RLD equation, 118–130 RHS. See right-hand side Richardson constant, 103 Richardson equation, 166 Richardson-Laue-Dushman (RLD), 45 Riemann zeta function, 9, 108, 152, 305–306 Right-hand side (RHS), 11 RLD. See Richardson-LaueDushman RLD equations, 104–106, 110–118 RPA. See Random-phase approximation RTD. See Resonant tunneling diode Runge-Kutta method, 246 Rydberg energy, 29, 185

S Scattering acoustic phonon, 208 electron-electron, 180–185 factor, 215–222

335 rates, 174–215, 255 charged impurity relaxation time, 177–179 electron-electron scattering, 180–185 electron-phonon scattering, 194–212 Fermi’s golden rule, 174–177 Matthiessen’s rule/specification of scattering terms, 212–215 monatomic linear chain of atoms, 186–194 number of sites, 179 sinusoidal potential, 185–186 terms, specification of, 212–215 Schematic representation, 52 Schottky emission cathodes, 105 Schottky-barrier-lowering, 222, 304 Schro¨dinger’s equation, 12, 15–16, 162 Schro¨dinger’s representations, 47, 49–52 Screened Coulomb potential, 15 SDDS. See Second-order downwind diVerence scheme Second order upwind diVerence scheme (SUDS), 242 Second-order downwind diVerence scheme (SDDS), 243 Semiconductors, 20, 179 Single barrier transmission probability, 70 Sintered tungsten dispenser cathode, 262 Sinusoidal potential, 185–186 SLAC. See Stanford Linear Accelerator Slater determinant, 25, 181 Slope factor ratio, 114 Small argument case, 79–80 Solid lead, 266 Sought-for linear dependence, 238 Sound velocity, 192, 196 Spatial Fourier transform, 226 Spatial inversion, 201

336 Specific heat capacity, 167 Specification of scattering terms, 212–215 Spindt quadratic approximation, 116 Spindt-type emitter, 128 Spin-orbit coupling, 12 Square barrier, 67–69 Stanford Linear Accelerator (SLAC), 147 Step function potential, 81 Stirling’s approximation, 132 Stupidity energy, 29 Submonolayer coverage, 285 SUDS. See Second order upwind diVerence scheme SuYciently high temperatures, 194 Surface barriers, 33–40 Surface eVects, 22–32 Surface of surface heating, 233 Surface self-diVusion, 129 Surface structure, 264–269

T Taylor expansion, 159, 169 Temperature calculated peak, 225 Debye, 208 excursions, 226 heat/corresponding, 225–227 liquid nitrogen, 218 maximum, 229 rise, 227–233 for tungsten surfaces, 249 suYciently high, 194 Temperature of laser-illuminated surface coupled heat equations, 234–235 diVusion of heat/corresponding temperature, 225–227 drive lasers, 222–223 electron-phonon coupling factor, 236–239 multiple pulses/temperature rise, 227–233

INDEX

photocathodes, 222–223 simple model of temperature increase due to laser pulse, 223–225 Temperature-dependent coeYcients, 247–253 TFA. See Thomas-Fermi approximation Theoretical energy distribution, 127 Theoretical intrinsic emittance, 278 Theoretical quantum eYciency model, 303 Thermal conductivity, 167–170, 214 Thermal emission, 47–146 current density, 47–64 in Bohm approach, 62–64 in classical distribution function approach, 47–49 in Gaussian potential barrier, 61–62 in Schro¨dinger/Heisenberg representations, 49–52 in Wigner distribution function approach, 52–62 exactly solvable models of, 65–85 numerical methods of, 94–101 numerical treatment of image charge potential, 95–99 numerical treatment of quadratic potential, 95 resonant tunneling, 99–102 recent revisions of standard thermal/field models in emission in thermal-field transition region revisited, 136–139 Forbes approach to evaluation of elliptical integrals, 131–136 revised FN-RLD equation/ inference of work function from experimental data in, 118–130 mixed thermal-field conditions, 123–126

337

INDEX

slope-intercept methods applied to field emission, 127–130 thermionic emission, 121–123 slope-intercept methods applied to, 127–130 thermal/equation, 102–118 emission equation integrals/their approximation, 106–110 Fowler-Nordheim/RichardsonLaue-Dushman equations, 104–106 triangular barrier of, 80–85 WKB area under curve models, 85–94 image charge barrier, 87–94 quadratic barrier, 85–86 Thermal emittance, 48, 143–146 Thermal equilibrium, 234, 235 Thermal-field equation, 139–146 completion of, 142 thermal emittance, 143–146 Thermal-field transition region, 136–139 Thermionic data, 292–296 Thermionic emission, 279 Thermionic emitter, 270 Thomas-Fermi approximation (TFA), 42 Thomas-Fermi parameter, 195 Three-step model, 219 Time acoustic phonon, 212 charged impurity relaxation, 177–179 dependent electric field, 157 electron-phonon relaxation, 238 independent shield Coulomb potential, 177 integration, 231 phonon relaxation, 212 Top-down perspective, 268 Total energy per unit volume, 29 Total relaxation time, 214 Trace space, 143

Transmission probability, 52, 68, 69, 101 Triangular barrier, 80–85 Triangular barrier emission probability, 85 Trivial multidimensional generalization, 198 Tungsten, 293, 296 argon-cleaned polycrystalline, 300 clean, emitter, 119 parameters, 223, 293 sintered, dispenser cathode, 262 temperature rise for, 249

U Ultraviolet (UV), 150 Uniform electron density, 31 UV. See Ultraviolet UV illumination, 153

W Wave function, 16, 51, 202 methodology airy function approach, 71–80 for constant potential segment, 65–67 large argument case, 78–79 multiple square barriers, 69–71 small argument case, 79–80 square barrier, 67–69 Wronskians of airy functions, 80 Wave packet spreading, 54–57 WDF. See Wigner distribution function approach Weighted areas, 284 Wentzel-Kramers-Brillouin (WKB), 62 Width, 35 Wiedemann-Franz Law, 170–171 Wigner distribution function approach (WDF), 52–62, 61

338 Wigner function, 158 Wigner trajectory case, 63 WKB. See Wentzel-KramersBrillouin Work function, 46, 284 with crystal face, 261–263 from experimental data, 118–130 reduction, Gyftopoulos-Levine model of, 286–292 Wronskians of airy functions, 80

INDEX

Y YLF. See Yttrium-lithium-fluoride Yttrium-lithium-fluoride (YLF), 147

Z Zero-temperature electron gas, 151